The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions and operations with polynomials, such as factoring polynomials. Factoring polynomials makes it easier to calculate outputs, simplify rational expressions, and solve equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). It notes that factoring polynomials is useful for evaluating polynomial expressions more easily, as demonstrated by an example evaluating the factored form of 2x3 - 5x2 + 2x for various values of x.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions and operations with polynomials, such as factoring polynomials. Factoring polynomials makes it easier to calculate outputs, simplify rational expressions, and solve equations. One example factors the polynomial 2x3 - 5x2 + 2x and shows it is easier to evaluate the factored form for different values of x than the original polynomial. The key purposes of factoring polynomials are to simplify calculations and operations.
The document discusses mathematical expressions and algebraic expressions. It provides examples of algebraic expressions like 3x^2 - 2x + 4 and explains how to perform operations on polynomial expressions, like factoring 64x^3 + 125 as (4x + 5)(16x^2 - 20x + 25). The key purposes of factoring polynomials are stated as making it easier to calculate outputs, simplify rational expressions, and solve equations. An example is given to evaluate the factored expression 2x^3 - 5x^2 + 2x for various values of x.
The document discusses complex numbers. It begins by explaining that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined such that i^2 = -1. A complex number is then defined as a number of the form a + bi, where a is the real part and bi is the imaginary part. Rules for adding, subtracting and multiplying complex numbers by treating i as a variable and setting i^2 to -1 are provided. Examples of solving equations and performing operations with complex numbers are given.
The document provides homework assignments and practice problems involving order of operations and evaluating expressions with exponents. It includes:
1) Assigning homework problems from the textbook pages 182 and 188 evaluating expressions and their divisibility.
2) Examples of evaluating expressions with exponents such as -x4 and (-x)4.
3) Practice problems simplifying expressions and evaluating expressions for given values using order of operations.
The document discusses expressions and polynomials. It provides examples of algebraic expressions and operations that can be performed on polynomials, such as factoring. Factoring polynomials is useful for easier evaluation, simplifying rational expressions, and solving equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). Factoring the polynomial 2x3 - 5x2 + 2x is recommended before evaluating it for specific values of x.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions involving variables and operations. Polynomial expressions are algebraic expressions that can be written in the form anxn + an-1xn-1 + ... + a1x + a0, where the ai coefficients are numbers. The document gives examples of factoring polynomials using formulas like a3b3 = (ab)(a2ab + b2). Factoring polynomials makes it easier to calculate outputs and simplify expressions for operations like addition and subtraction.
47 operations of 2nd degree expressions and formulasalg1testreview
ย
The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.
The document discusses expressions and equations. It provides examples of using expressions to calculate total costs given individual costs, and using equations to solve for unknown variables. Specifically, it gives an example of calculating the total cost of x pizzas using the expression "8x + 10" and solving the equation "8x + 10 = 810" to determine that x = 100 pizzas were ordered. It then discusses how to solve rational equations by clearing denominators using the lowest common denominator. An example problem demonstrates solving the rational equation (x - 2)/(x + 1) = 2/4 + 1 through multiplying both sides by the LCD (x - 2)(x + 1).
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions and operations with polynomials, such as factoring polynomials. Factoring polynomials makes it easier to calculate outputs, simplify rational expressions, and solve equations. One example factors the polynomial 2x3 - 5x2 + 2x and shows it is easier to evaluate the factored form for different values of x than the original polynomial. The key purposes of factoring polynomials are to simplify calculations and operations.
The document discusses mathematical expressions and algebraic expressions. It provides examples of algebraic expressions like 3x^2 - 2x + 4 and explains how to perform operations on polynomial expressions, like factoring 64x^3 + 125 as (4x + 5)(16x^2 - 20x + 25). The key purposes of factoring polynomials are stated as making it easier to calculate outputs, simplify rational expressions, and solve equations. An example is given to evaluate the factored expression 2x^3 - 5x^2 + 2x for various values of x.
The document discusses complex numbers. It begins by explaining that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined such that i^2 = -1. A complex number is then defined as a number of the form a + bi, where a is the real part and bi is the imaginary part. Rules for adding, subtracting and multiplying complex numbers by treating i as a variable and setting i^2 to -1 are provided. Examples of solving equations and performing operations with complex numbers are given.
The document provides homework assignments and practice problems involving order of operations and evaluating expressions with exponents. It includes:
1) Assigning homework problems from the textbook pages 182 and 188 evaluating expressions and their divisibility.
2) Examples of evaluating expressions with exponents such as -x4 and (-x)4.
3) Practice problems simplifying expressions and evaluating expressions for given values using order of operations.
The document discusses expressions and polynomials. It provides examples of algebraic expressions and operations that can be performed on polynomials, such as factoring. Factoring polynomials is useful for easier evaluation, simplifying rational expressions, and solving equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). Factoring the polynomial 2x3 - 5x2 + 2x is recommended before evaluating it for specific values of x.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions involving variables and operations. Polynomial expressions are algebraic expressions that can be written in the form anxn + an-1xn-1 + ... + a1x + a0, where the ai coefficients are numbers. The document gives examples of factoring polynomials using formulas like a3b3 = (ab)(a2ab + b2). Factoring polynomials makes it easier to calculate outputs and simplify expressions for operations like addition and subtraction.
47 operations of 2nd degree expressions and formulasalg1testreview
ย
The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.
The document discusses expressions and equations. It provides examples of using expressions to calculate total costs given individual costs, and using equations to solve for unknown variables. Specifically, it gives an example of calculating the total cost of x pizzas using the expression "8x + 10" and solving the equation "8x + 10 = 810" to determine that x = 100 pizzas were ordered. It then discusses how to solve rational equations by clearing denominators using the lowest common denominator. An example problem demonstrates solving the rational equation (x - 2)/(x + 1) = 2/4 + 1 through multiplying both sides by the LCD (x - 2)(x + 1).
1.0 factoring trinomials the ac method and making lists-tmath260tutor
ย
This document contains examples and explanations of factoring trinomials using various methods:
1. The ac-method is used to factor 3x^2 - 4x - 20 by finding numbers that satisfy uv = ac and u + v = b.
2. Another trinomial, 3x^2 - 6x - 20, is shown to be prime because its ac-list does not contain numbers satisfying the required equations.
3. The method of checking if b^2 - 4ac is a perfect square is demonstrated, indicating whether a trinomial is factorable or prime.
The document discusses sign charts for factorable formulas. It provides examples of determining the sign (positive or negative) of expressions when evaluated at given values of x by factoring the expressions into their factored forms. The key steps to create a sign chart are: 1) solve for values where the expression is equal to 0, 2) mark these values on a number line, 3) select points in each segment to test the sign, 4) indicate the sign (positive or negative) in each segment based on the testing. Sign charts show the regions where an expression is positive, negative or equal to 0.
1.0 factoring trinomials the ac method and making lists-xmath260
ย
The document discusses factoring trinomials and making lists of numbers to help determine which trinomials are factorable. It states that trinomials are either factorable, where they can be written as the product of two binomials, or prime/unfactorable. Making lists of numbers that satisfy certain criteria, like having a product of the top number in a table, can help identify factorable trinomials and determine the factors.
The document discusses using sign charts to determine the sign (positive, negative, or zero) of polynomials and rational expressions for different values of x. It provides examples of drawing sign charts for various expressions and using them to solve inequality statements. Key steps include factoring expressions, identifying zeros and undefined values, and testing sample points in each interval to determine the sign over that interval. Sign charts can then be used to easily solve inequality statements by identifying the intervals where the expression is positive or negative.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
The document discusses different methods for solving equations, including:
- Solving 1st and 2nd degree polynomial equations by setting them equal to 0 and using factoring or the quadratic formula.
- Solving rational equations by clearing all denominators using the lowest common denominator.
- Solving equations may require transforming them into polynomial equations first through methods like factoring or factoring by grouping.
The document discusses operations that can be performed on polynomial expressions. It defines terms and like-terms in polynomials, and explains that like-terms can be combined while unlike terms cannot. It provides examples of combining like-terms, expanding polynomials using the distributive property, multiplying terms and polynomials, and simplifying the results.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
This document provides examples and explanations of operations involving polynomials and rational expressions. It covers factoring polynomials, evaluating polynomial expressions, adding, subtracting, multiplying and dividing rational expressions, and simplifying complex fractions and expressions with radicals. Step-by-step solutions are shown for problems such as factoring expressions, evaluating polynomials for given values, combining like rational expressions, rationalizing denominators, and more. The document demonstrates various techniques for working with polynomials and rational expressions.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
1 s2 addition and subtraction of signed numbersmath123a
ย
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
The document discusses order of operations and provides examples to illustrate how to correctly evaluate mathematical expressions involving multiple operations. It establishes that the order of operations is: 1) operations within grouping symbols from innermost to outermost, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Examples with step-by-step workings demonstrate applying this order of operations to evaluate expressions involving grouping symbols, multiplication, division, addition and subtraction.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses methods for multiplying binomial expressions. It defines a binomial as a two-term polynomial of the form ax + b, and a trinomial as a three-term polynomial of the form ax^2 + bx + c. It then introduces the FOIL method for multiplying two binomials, which results in a trinomial. FOIL stands for First, Outer, Inner, Last - referring to which terms to multiply to obtain each term in the trinomial product. The document provides examples working through multiplying binomials using FOIL. It also discusses approaches for expanding expressions with a negative binomial, such as distributing the negative sign first before using FOIL.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
The document discusses sign charts for factorable formulas. It provides examples of drawing sign charts, including marking the roots of factors and determining the sign of the expression between roots based on whether roots are odd- or even-ordered. Exercises have students draw sign charts for additional formulas and use them to solve inequalities or determine domains where expressions are defined.
The document provides information about polynomials at Higher level, including:
- Definitions of polynomials and examples of polynomial expressions
- Evaluating polynomials using substitution and nested/synthetic methods
- The factor theorem and using it to factorize polynomials
- Finding missing coefficients in polynomials
- Finding the polynomial expression given its zeros
To summarize the key steps for factoring polynomials:
1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
3. Integer roots that produce a zero remainder are factors of the polynomial. Repeating this process allows one to fully factor the polynomial into linear terms.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
1. The document discusses dividing polynomials by using long division or writing an identity and equating coefficients.
2. It provides examples of using both methods to divide polynomials and determine quotients and remainders.
3. The Remainder Theorem is introduced, which states that when a polynomial f(x) is divided by (x - a), the remainder is equal to the value of f(a).
This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.
This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.
1.0 factoring trinomials the ac method and making lists-tmath260tutor
ย
This document contains examples and explanations of factoring trinomials using various methods:
1. The ac-method is used to factor 3x^2 - 4x - 20 by finding numbers that satisfy uv = ac and u + v = b.
2. Another trinomial, 3x^2 - 6x - 20, is shown to be prime because its ac-list does not contain numbers satisfying the required equations.
3. The method of checking if b^2 - 4ac is a perfect square is demonstrated, indicating whether a trinomial is factorable or prime.
The document discusses sign charts for factorable formulas. It provides examples of determining the sign (positive or negative) of expressions when evaluated at given values of x by factoring the expressions into their factored forms. The key steps to create a sign chart are: 1) solve for values where the expression is equal to 0, 2) mark these values on a number line, 3) select points in each segment to test the sign, 4) indicate the sign (positive or negative) in each segment based on the testing. Sign charts show the regions where an expression is positive, negative or equal to 0.
1.0 factoring trinomials the ac method and making lists-xmath260
ย
The document discusses factoring trinomials and making lists of numbers to help determine which trinomials are factorable. It states that trinomials are either factorable, where they can be written as the product of two binomials, or prime/unfactorable. Making lists of numbers that satisfy certain criteria, like having a product of the top number in a table, can help identify factorable trinomials and determine the factors.
The document discusses using sign charts to determine the sign (positive, negative, or zero) of polynomials and rational expressions for different values of x. It provides examples of drawing sign charts for various expressions and using them to solve inequality statements. Key steps include factoring expressions, identifying zeros and undefined values, and testing sample points in each interval to determine the sign over that interval. Sign charts can then be used to easily solve inequality statements by identifying the intervals where the expression is positive or negative.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
The document discusses different methods for solving equations, including:
- Solving 1st and 2nd degree polynomial equations by setting them equal to 0 and using factoring or the quadratic formula.
- Solving rational equations by clearing all denominators using the lowest common denominator.
- Solving equations may require transforming them into polynomial equations first through methods like factoring or factoring by grouping.
The document discusses operations that can be performed on polynomial expressions. It defines terms and like-terms in polynomials, and explains that like-terms can be combined while unlike terms cannot. It provides examples of combining like-terms, expanding polynomials using the distributive property, multiplying terms and polynomials, and simplifying the results.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
This document provides examples and explanations of operations involving polynomials and rational expressions. It covers factoring polynomials, evaluating polynomial expressions, adding, subtracting, multiplying and dividing rational expressions, and simplifying complex fractions and expressions with radicals. Step-by-step solutions are shown for problems such as factoring expressions, evaluating polynomials for given values, combining like rational expressions, rationalizing denominators, and more. The document demonstrates various techniques for working with polynomials and rational expressions.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
1 s2 addition and subtraction of signed numbersmath123a
ย
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
The document discusses order of operations and provides examples to illustrate how to correctly evaluate mathematical expressions involving multiple operations. It establishes that the order of operations is: 1) operations within grouping symbols from innermost to outermost, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Examples with step-by-step workings demonstrate applying this order of operations to evaluate expressions involving grouping symbols, multiplication, division, addition and subtraction.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses methods for multiplying binomial expressions. It defines a binomial as a two-term polynomial of the form ax + b, and a trinomial as a three-term polynomial of the form ax^2 + bx + c. It then introduces the FOIL method for multiplying two binomials, which results in a trinomial. FOIL stands for First, Outer, Inner, Last - referring to which terms to multiply to obtain each term in the trinomial product. The document provides examples working through multiplying binomials using FOIL. It also discusses approaches for expanding expressions with a negative binomial, such as distributing the negative sign first before using FOIL.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
The document discusses sign charts for factorable formulas. It provides examples of drawing sign charts, including marking the roots of factors and determining the sign of the expression between roots based on whether roots are odd- or even-ordered. Exercises have students draw sign charts for additional formulas and use them to solve inequalities or determine domains where expressions are defined.
The document provides information about polynomials at Higher level, including:
- Definitions of polynomials and examples of polynomial expressions
- Evaluating polynomials using substitution and nested/synthetic methods
- The factor theorem and using it to factorize polynomials
- Finding missing coefficients in polynomials
- Finding the polynomial expression given its zeros
To summarize the key steps for factoring polynomials:
1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
3. Integer roots that produce a zero remainder are factors of the polynomial. Repeating this process allows one to fully factor the polynomial into linear terms.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
1. The document discusses dividing polynomials by using long division or writing an identity and equating coefficients.
2. It provides examples of using both methods to divide polynomials and determine quotients and remainders.
3. The Remainder Theorem is introduced, which states that when a polynomial f(x) is divided by (x - a), the remainder is equal to the value of f(a).
This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.
This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.
This document defines polynomials and discusses their key properties. A polynomial is an expression of the form anxn + an-1xn-1 + ... + a1x + a0, where the coefficients an, an-1, ..., a1, a0 are real numbers. The degree of a polynomial is the highest exponent in the expression, and the leading coefficient and term refer to the coefficient and term with the highest degree. Addition and multiplication of polynomials follow the distributive property, and the degree and leading term of a product are determined by the individual polynomials' degrees and leading terms.
Real numbers include rational numbers like fractions and irrational numbers like square roots. Real numbers are represented by the symbol R. They consist of natural numbers, whole numbers, integers, rational numbers and irrational numbers. [/SUMMARY]
Project in math BY:Samuel Vasquez Baliasamuel balia
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Real numbers include rational numbers like fractions as well as irrational numbers like the square root of 2. Real numbers are represented by the symbol R and include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as fractions with integers as the numerator and non-zero denominator, while irrational numbers cannot be expressed as fractions.
This document discusses algebraic expressions and how to work with them. It defines an expression as a symbol or combination of symbols that represents a value or relation. It explains how to write verbal and symbolic expressions, calculate the value of an expression by plugging in numbers, and factor expressions by finding the greatest common factor or grouping terms. Examples are provided for each concept to illustrate the process. Resources for further information on algebraic expressions are listed at the end.
The document discusses polynomials and polynomial functions. It defines a polynomial as a sum of monomials, with a monomial being a variable or the product of a variable and real numbers with whole number exponents. It classifies polynomials by degree and number of terms, with examples of common types like linear, quadratic, and cubic polynomials. It also defines a polynomial function as a function represented by a polynomial, and discusses finding sums, differences, and writing polynomials in standard form.
Algebraic expressions can be written in verbal, symbolic, or calculated forms. Verbal expressions use words to represent variables and relations, while symbolic expressions use symbols. To calculate an expression, values are plugged in for variables and the expression is evaluated. Expressions can also be factored by finding a greatest common factor and grouping like terms.
Algebraic expressions can be written in verbal, symbolic, or calculated forms. Verbal expressions use words to represent variables and relations, while symbolic expressions use symbols. To calculate an expression, values are plugged in for the variables and the expression is evaluated. Expressions can also be factored by finding a greatest common factor and grouping like terms.
1) The document discusses various methods for manipulating and solving algebraic expressions, including adding, subtracting, and factoring polynomials.
2) Factoring techniques include grouping like terms, using the difference of squares formula, and recognizing perfect square trinomials.
3) The quadratic formula is introduced as a way to solve quadratic equations of the form ax2 + bx + c = 0.
The document discusses various properties of real numbers including the commutative, associative, identity, inverse, zero, and distributive properties. It also covers topics such as combining like terms, translating word phrases to algebraic expressions, and simplifying algebraic expressions. Examples are provided to illustrate each concept along with explanations of key terms like coefficients, variables, and like terms.
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.
1. The document shows methods for calculating the area of rectangles by splitting them into smaller rectangles.
2. It demonstrates that the area of the original rectangle equals the sum of the areas of the smaller rectangles.
3. Algebraic formulas are developed to represent splitting rectangles and multiplying sums and differences.
This document provides information on various mathematical concepts including:
1. A monomial is the product of a real number and one or more variables. A polynomial contains more than one variable, constants, and exponents.
2. When adding or subtracting polynomials, like terms are combined. When multiplying polynomials, each term in one polynomial is multiplied by each term in the other and the results are added.
3. Fractions can be added by finding a common denominator and adding the numerators. Fractions are multiplied by multiplying the numerators and multiplying the denominators.
4. Equations can be solved using techniques like reduction, substitution, and factoring polynomials.
5. Factoring is the process of writing an
Factoring polynomials involves finding common factors that can be divided out of terms, similar to factoring numbers but with variables; this is done by looking for a single variable or number that is a common factor of all terms that can be pulled out in front of parentheses. The document provides examples of different types of factoring polynomials including using the greatest common factor, difference of squares, grouping, and perfect squares and cubes.
Maths ppt on algebraic expressions and identitesANKIT SAHOO
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This document discusses algebraic expressions and identities. It defines expressions as combinations of numbers and variables connected by operation signs. Expressions can be monomials containing one term, binomials containing two terms, or trinomials containing three terms. Terms are separated parts of expressions and factors are the numbers within terms. Coefficients are factors without signs. The document also covers adding, subtracting and multiplying expressions, as well as defining identities as equalities that are true for all variable values. It provides examples of standard identities for the sum and difference of squares and multiplying the sum and difference of two terms.
factoring trinomials the ac method and making listsmath260
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This document discusses factoring trinomials. It defines a trinomial as a three-term polynomial of the form ax2 + bx + c, where a, b, and c are numbers. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are prime or unfactorable. The document outlines the basic rules for factoring trinomials and provides an example of expanding a binomial expression into a trinomial.
The document provides instructions on factorizing quadratic equations. It begins by explaining what quadratic equations are and provides examples. It then discusses factorizing quadratics where the coefficient of x^2 is 1 by finding two numbers whose product is the last term and sum is the middle term. The document continues explaining how to factorize when the coefficient of x^2 is not 1 and predicts the signs of the factors based on the signs of the terms in the quadratic equation. It provides examples of factorizing different quadratic equations.
Polynomials are algebraic expressions involving variables and their powers. The degree of a polynomial is the highest power of the variable. To add or subtract polynomials, like terms must be combined. To multiply polynomials, the distributive property and FOIL method are used. Special formulas exist for multiplying the sum and difference of expressions.
The document introduces matrices and matrix operations. Matrices are rectangular tables of numbers that are used for applications beyond solving systems of equations. Matrix notation defines a matrix with R rows and C columns as an R x C matrix. The entry in the ith row and jth column is denoted as aij. Matrices can be added or subtracted if they are the same size by adding or subtracting the corresponding entries. There are two types of matrix multiplication: scalar multiplication multiplies a matrix by a constant, and matrix multiplication involves multiplying corresponding rows and columns where the number of columns of the left matrix equals the rows of the right matrix.
35 Special Cases System of Linear Equations-x.pptxmath260
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The document discusses special cases of systems of linear equations, including inconsistent/contradictory systems where the equations are impossible to satisfy simultaneously, and dependent systems where there are infinitely many solutions. An inconsistent system is shown with equations x + y = 2 and x + y = 3, which has no solution since they cannot both be true. A dependent system is shown with equations x + y = 2 and 2x + 2y = 4, which has infinitely many solutions like (2,0) and (1,1). The row-reduced echelon form (rref) of a matrix is also discussed, which puts a system of equations in a standard form to help determine if it is consistent, dependent, or has
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
19 more parabolas a& hyperbolas (optional) xmath260
ย
After dividing the general quadratic equation Ax2 + By2 + Cx + Dy = E by A, three types of conic sections can be obtained:
1) Parabolas occur when B = 0, resulting in equations of the form 1x2 + #x + #y = #.
2) Circles occur when A = B, resulting in the equation 1x2 + 1y2 = 1.
3) Hyperbolas occur when A and B have opposite signs, resulting in equations of the form 1x2 + ry2 + #x + #y = # with r < 0. Hyperbolas have two foci and asymptotes, and points on the hyperbola have
The document discusses conic sections, specifically circles and ellipses. It defines an ellipse as the set of points where the sum of the distances to two fixed foci is a constant. An ellipse has a center, two axes (semi-major and semi-minor), and can be represented by the standard form (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. Examples are provided to demonstrate finding attributes of ellipses from their equations.
This document discusses conic sections and first degree equations. It begins by introducing conic sections as the shapes formed by slicing a cone at different angles. It then covers first degree equations, noting that their graphs are straight lines that can be written in the form of y=mx+b. Specific examples of first degree equations and their graphs are shown. The document ends by introducing the four types of conic sections - circles, ellipses, parabolas, and hyperbolas - and how graphs of second degree equations can represent these shapes.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
14 graphs of factorable rational functions xmath260
ย
The document discusses graphs of rational functions. It defines rational functions as functions of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials. It describes how vertical asymptotes occur where the denominator Q(x) is zero. The graph runs along either side of vertical asymptotes, going up or down depending on the sign chart. There are four cases for how the graph behaves at a vertical asymptote. The document uses examples to illustrate graphing rational functions and determining vertical asymptotes. It also mentions horizontal asymptotes will be discussed.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a โ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses the basic language of functions. It defines a function as a procedure that assigns each input exactly one output. Functions can be represented by formulas using typical variables like f(x) = x^2 - 2x + 3, where x is the input and f(x) is the output. Functions have a domain, which is the set of all possible inputs, and a range, which is the set of all possible outputs. Functions can be depicted graphically or via tables listing inputs and outputs.
The document discusses inverse functions. An inverse function reverses the input and output of a function. For a function f(x) to have an inverse function f^-1(y), it must be one-to-one, meaning that different inputs map to different outputs. The inverse of f(x) is obtained by solving the original function equation for x in terms of y. Examples show how to determine if a function has an inverse and how to calculate the inverse function. For non one-to-one functions like f(x)=x^2, the inverse procedure is not a well-defined function.
This document discusses two types of log and exponential equations: those that do not require calculators and numerical equations that do require calculators. Equations that do not require calculators can be solved by putting both sides into a common base, consolidating exponents, and dropping the base to solve the resulting equation. For log equations, logs are consolidated on each side first before dropping the log. Two examples demonstrating these solution methods are provided.
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving both log equations and exponential equations. To solve log equations, the problems are rewritten in exponential form by removing the log. To solve exponential equations, the exponents are brought down by rewriting the problem in logarithmic form. Steps include isolating the exponential or logarithmic term containing the unknown, rewriting the equation accordingly, and then solving for the unknown variable. Practice problems with solutions are provided to illustrate the process.
The document discusses exponential and logarithmic expressions. Exponential expressions like 43, 82, 26 all equal 64. Their corresponding logarithmic forms are log4(64), log8(64), log2(64) and equal 3, 2, 6 respectively. When working with exponential or logarithmic expressions, the base number must be identified first. Both numbers in the logarithmic expression logb(y) must be positive.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
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Ivรกn Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
ย
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
How Barcodes Can Be Leveraged Within Odoo 17Celine George
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In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
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These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
ย
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
ย
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
3. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Expressions
4. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
Expressions
5. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula โ8x + 10โ.
Expressions
6. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula โ8x + 10โ.
Such a formula is called an expression.
Expressions
7. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula โ8x + 10โ.
Such a formula is called an expression.
Expressions
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810.
8. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula โ8x + 10โ.
Such a formula is called an expression.
Expressions
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810. The value x = 100 is called the
input and the projected cost $810 is called the output.
9. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula โ8x + 10โ.
Such a formula is called an expression.
Expressions
Definition: Mathematical expressions are calculation
procedures which are written with numbers, variables,
operation symbols +, โ, *, / and ( )โs.
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810. The value x = 100 is called the
input and the projected cost $810 is called the output.
10. We order pizzas from Pizza Grande.
Each pizza is $8 and there is a $10 delivery charge.
Hence if we ordered 5 pizzas delivered, the total cost
would be 8(5) + 10 = $50, excluding the tip.
If we want x pizzas delivered, then the total cost is
given by the formula โ8x + 10โ.
Such a formula is called an expression.
Expressions
Definition: Mathematical expressions are calculation
procedures which are written with numbers, variables,
operation symbols +, โ, *, / and ( )โs.
Expressions calculate the expected future results.
If we ordered x = 100 pizzas, the cost would be
8(100)+10 = $810. The value x = 100 is called the
input and the projected cost $810 is called the output.
11. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
12. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Trigonometric or log-formulas
are not algebraic.
13. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 โ 2x + 4,
14. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 โ 2x + 4,
x2 + 3
๏3 x3 โ 2x โ 4
,
15. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 โ 2x + 4,
x2 + 3
๏3 x3 โ 2x โ 4
,
(x1/2 + y)1/3
(4y2 โ (x + 4)1/2)1/4
16. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 โ 2x + 4,
x2 + 3
๏3 x3 โ 2x โ 4
,
(x1/2 + y)1/3
(4y2 โ (x + 4)1/2)1/4
Examples of nonโalgebraic expressions are
sin(x), 2x, log(x + 1).
17. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 โ 2x + 4,
x2 + 3
๏3 x3 โ 2x โ 4
,
(x1/2 + y)1/3
(4y2 โ (x + 4)1/2)1/4
Examples of nonโalgebraic expressions are
sin(x), 2x, log(x + 1).
The algebraic expressions anxn + anโ1xnโ1...+ a1x + a0
where ai are numbers, are called polynomials (in x).
18. An algebraic expression is a formula constructed
with variables and numbers using addition,
subtraction, multiplication, division, and taking roots.
Algebraic Expressions
Examples of algebraic expressions are
3x2 โ 2x + 4,
x2 + 3
๏3 x3 โ 2x โ 4
,
(x1/2 + y)1/3
(4y2 โ (x + 4)1/2)1/4
Examples of nonโalgebraic expressions are
sin(x), 2x, log(x + 1).
The algebraic expressions anxn + anโ1xnโ1...+ a1x + a0
where ai are numbers, are called polynomials (in x).
The algebraic expressions where P and Q are
polynomials, are called rational expressions.
P
Q
20. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ (3x โ 4)(x + 5)
21. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ (3x โ 4)(x + 5)
The point of this problem is
how to subtract a โproductโ.
22. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)] Insert [ ]
The point of this problem is
how to subtract a โproductโ.
23. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
Insert [ ]
remove [ ]
24. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
= 2x2 + x โ 15 โ 3x2 โ 11x + 20
Insert [ ]
remove [ ]
25. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
= 2x2 + x โ 15 โ 3x2 โ 11x + 20
= โx2 โ 10x + 5
Insert [ ]
remove [ ]
26. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
= 2x2 + x โ 15 โ 3x2 โ 11x + 20
= โx2 โ 10x + 5
Insert [ ]
remove [ ]
27. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
= 2x2 + x โ 15 โ 3x2 โ 11x + 20
= โx2 โ 10x + 5
Insert [ ]
remove [ ]
(2x โ 5)(x +3) โ (3x โ 4)(x + 5)
= (2x โ 5)(x +3) + (โ3x + 4)(x + 5)
= โฆ
Or distribute the minus sign and
change it to an addition problem:
28. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
= 2x2 + x โ 15 โ 3x2 โ 11x + 20
= โx2 โ 10x + 5
Insert [ ]
To factor an expression
means to write it as a product
in a non-obvious way.
remove [ ]
(2x โ 5)(x +3) โ (3x โ 4)(x + 5)
= (2x โ 5)(x +3) + (โ3x + 4)(x + 5)
= โฆ
Or distribute the minus sign and
change it to an addition problem:
29. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x โ 5)(x +3) โ [(3x โ 4)(x + 5)]
= 2x2 + x โ 15 โ [3x2 + 11x โ 20]
= 2x2 + x โ 15 โ 3x2 โ 11x + 20
= โx2 โ 10x + 5
Insert [ ]
A3 B3 = (A B)(A2 AB + B2)
Important Factoring Formulas:
To factor an expression
means to write it as a product
in a non-obvious way.
A2 โ B2 = (A + B)(A โ B)
+
โ
+
โ +
โ
remove [ ]
(2x โ 5)(x +3) โ (3x โ 4)(x + 5)
= (2x โ 5)(x +3) + (โ3x + 4)(x + 5)
= โฆ
Or distribute the minus sign and
change it to an addition problem:
30. Example B. Factor 64x3 + 125
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
31. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
A3 B3
32. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 โ (4x)(5) +(5)2)
Polynomial Expressions
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
(A B) (A2 AB + B2)
+ โ
34. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 โ (4x)(5) +(5)2)
= (4x + 5)(16x2 โ 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
35. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 โ (4x)(5) +(5)2)
= (4x + 5)(16x2 โ 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. Itโs easier to calculate an output or to check
the sign of an output using the factored form.
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
36. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 โ (4x)(5) +(5)2)
= (4x + 5)(16x2 โ 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. Itโs easier to calculate an output or to check
the sign of an output using the factored form.
II. To simplify or perform algebraic operations with
rational expressions.
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
37. Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 โ (4x)(5) +(5)2)
= (4x + 5)(16x2 โ 20x + 25)
Polynomial Expressions
We factor polynomials for the following purposes.
I. Itโs easier to calculate an output or to check
the sign of an output using the factored form.
II. To simplify or perform algebraic operations with
rational expressions.
III. To solve equations (See next section).
A3 B3 = (A B)(A2 AB + B2)
+
โ +
โ
+
โ
38. Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions.
Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
39. Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
40. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
41. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
2x3 โ 5x2 + 2x = x(2x2 โ 5x + 2)
= x(2x โ 1)(x โ 2)
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
42. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
2x3 โ 5x2 + 2x = x(2x2 โ 5x + 2)
= x(2x โ 1)(x โ 2)
Plug in x = โ2:
โ2 [2(โ2) โ 1] [(โ2) โ 2]
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
43. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
2x3 โ 5x2 + 2x = x(2x2 โ 5x + 2)
= x(2x โ 1)(x โ 2)
Plug in x = โ2:
โ2 [2(โ2) โ 1] [(โ2) โ 2] = โ2 [โ5] [โ4] = โ40
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
44. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
2x3 โ 5x2 + 2x = x(2x2 โ 5x + 2)
= x(2x โ 1)(x โ 2)
Plug in x = โ2:
โ2 [2(โ2) โ 1] [(โ2) โ 2] = โ2 [โ5] [โ4] = โ40
Plug in x = โ1:
โ1 [2(โ1) โ 1] [(โ1) โ 2]
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
45. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
2x3 โ 5x2 + 2x = x(2x2 โ 5x + 2)
= x(2x โ 1)(x โ 2)
Plug in x = โ2:
โ2 [2(โ2) โ 1] [(โ2) โ 2] = โ2 [โ5] [โ4] = โ40
Plug in x = โ1:
โ1 [2(โ1) โ 1] [(โ1) โ 2] = โ1 [โ3] [โ3] = โ9
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
46. Example C. Evaluate 2x3 โ 5x2 + 2x for x = โ2, โ1, 3
by factoring it first.
2x3 โ 5x2 + 2x = x(2x2 โ 5x + 2)
= x(2x โ 1)(x โ 2)
Plug in x = โ2:
โ2 [2(โ2) โ 1] [(โ2) โ 2] = โ2 [โ5] [โ4] = โ40
Plug in x = โ1:
โ1 [2(โ1) โ 1] [(โ1) โ 2] = โ1 [โ3] [โ3] = โ9
Plug in x = 3:
3 [2(3) โ 1] [(3) โ 2] = 3 [5] [1] = 15
Evaluate Polynomial Expressions
It's easier to evaluate factored polynomial
expressions. It takes fewer steps then plugging in
the values directly.
47. Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
48. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
49. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
x2 โ 2x โ 3 = (x โ 3)(x + 1).
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
50. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
x2 โ 2x โ 3 = (x โ 3)(x + 1). Hence for x = โ3/2,
we get (โ3/2 โ 3)(โ3/2 + 1)
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
51. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
x2 โ 2x โ 3 = (x โ 3)(x + 1). Hence for x = โ3/2,
we get (โ3/2 โ 3)(โ3/2 + 1) is (โ)(โ) = + .
Determine the Signs of the Outputs.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
52. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
x2 โ 2x โ 3 = (x โ 3)(x + 1). Hence for x = โ3/2,
we get (โ3/2 โ 3)(โ3/2 + 1) is (โ)(โ) = + .
Determine the Signs of the Outputs.
Rational Expressions
We say a rational expression is in the factored form
if it's numerator and denominator are factored.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
53. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
x2 โ 2x โ 3 = (x โ 3)(x + 1). Hence for x = โ3/2,
we get (โ3/2 โ 3)(โ3/2 + 1) is (โ)(โ) = + .
Determine the Signs of the Outputs.
Rational Expressions
We say a rational expression is in the factored form
if it's numerator and denominator are factored.
Example E. Factor
x2 โ 1
x2 โ 3x+ 2
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
54. Example D. Determine whether the outcome is + or โ
for x2 โ 2x โ 3 if x = โ3/2.
x2 โ 2x โ 3 = (x โ 3)(x + 1). Hence for x = โ3/2,
we get (โ3/2 โ 3)(โ3/2 + 1) is (โ)(โ) = + .
Determine the Signs of the Outputs.
Rational Expressions
We say a rational expression is in the factored form
if it's numerator and denominator are factored.
Example E. Factor
x2 โ 1
x2 โ 3x+ 2
x2 โ 1
x2 โ 3x+ 2
=
(x โ 1)(x + 1)
(x โ 1)(x โ 2)
is the factored form.
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
55. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
56. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.
x*y
x*z =
x*y
x*z =
y
z
1
57. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.
58. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Example F. Reduce x2 โ 1
x2 โ 3x+ 2
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.
59. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Example F. Reduce x2 โ 1
x2 โ 3x+ 2
x2 โ 1
x2 โ 3x+ 2 =
(x โ 1)(x + 1)
(x โ 1)(x โ 2)
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
factor
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.
60. Rational Expressions
We put rational expressions in the factored form
in order to reduce, multiply or divide them.
Example F. Reduce x2 โ 1
x2 โ 3x+ 2
x2 โ 1
x2 โ 3x+ 2 =
(x โ 1)(x + 1)
(x โ 1)(x โ 2)
x*y
x*z =
x*y
x*z =
y
z
A rational expression that can't be cancelled any
further is said to be reduced.
= (x + 1)
(x โ 2)
factor
Cancellation Rule: Given a rational expression in
the factored form, common factors may be cancelled,
i.e.
66. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
รท = P*S
Q*R
Reciprocate
Example G. Simplify (2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
(2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
=
(2x โ 6)
(y + 3)
(y2 + 2y โ 3)
(9 โ x2)
*
Reciprocate
67. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
รท = P*S
Q*R
Reciprocate
Example G. Simplify (2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
(2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
=
(2x โ 6)
(y + 3)
(y2 + 2y โ 3)
(9 โ x2)
*
Reciprocate
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
68. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
รท = P*S
Q*R
Reciprocate
Example G. Simplify (2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
(2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
=
(2x โ 6)
(y + 3)
(y2 + 2y โ 3)
(9 โ x2)
*
=
2(x โ 3)
(y + 3)
(y + 3)(y โ 1)
(3 โ x)(3 + x)
* factor and cancel
69. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
รท = P*S
Q*R
Reciprocate
Example G. Simplify (2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
(2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
=
(2x โ 6)
(y + 3)
(y2 + 2y โ 3)
(9 โ x2)
*
=
2(x โ 3)
(y + 3)
(y + 3)(y โ 1)
(3 โ x)(3 + x)
*
1
factor and cancel
70. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
รท = P*S
Q*R
Reciprocate
Example G. Simplify (2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
(2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
=
(2x โ 6)
(y + 3)
(y2 + 2y โ 3)
(9 โ x2)
*
=
2(x โ 3)
(y + 3)
(y + 3)(y โ 1)
(3 โ x)(3 + x)
*
โ1 1
factor and cancel
71. Rational Expressions
Multiplication Rule:
To carry out these operations, put the expressions
in factored form and cancel as much as possible.
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
รท = P*S
Q*R
Reciprocate
Example G. Simplify (2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
(2x โ 6)
(y + 3) รท
(y2 + 2y โ 3)
(9 โ x2)
=
(2x โ 6)
(y + 3)
(y2 + 2y โ 3)
(9 โ x2)
*
=
2(x โ 3)
(y + 3)
(y + 3)(y โ 1)
(3 โ x)(3 + x)
*
โ1 1
=
โ2(y โ 1)
(x + 3)
73. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
II. to simplify complex fractions
74. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
II. to simplify complex fractions
III. to solve rational equations (later)
75. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
II. to simplify complex fractions
III. to solve rational equations (later)
76. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
Combining Rational Expressions (LCD Method):
II. to simplify complex fractions
III. to solve rational equations (later)
77. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD,
II. to simplify complex fractions
III. to solve rational equations (later)
78. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48,
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD,
II. to simplify complex fractions
III. to solve rational equations (later)
79. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
7
12
5
8
+ โ
16
9
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD,
48
II. to simplify complex fractions
III. to solve rational equations (later)
80. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
7
12
5
8
+ โ
16
9 48
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD, expand (F ยฑ G)* LCD
and simplify (F ยฑ G)(LCD) / LCD.
II. to simplify complex fractions
III. to solve rational equations (later)
81. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
7
12
5
8
+ โ
16
9
4
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD, expand (F ยฑ G)* LCD
and simplify (F ยฑ G)(LCD) / LCD.
48
II. to simplify complex fractions
III. to solve rational equations (later)
82. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
6
7
12
5
8
+ โ
16
9
4
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD, expand (F ยฑ G)* LCD
and simplify (F ยฑ G)(LCD) / LCD.
48
II. to simplify complex fractions
III. to solve rational equations (later)
83. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
6
7
12
5
8
+ โ
16
9
4 3
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD, expand (F ยฑ G)* LCD
and simplify (F ยฑ G)(LCD) / LCD.
48
II. to simplify complex fractions
III. to solve rational equations (later)
84. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
6
7
12
5
8
+ โ
16
9
4 3
=
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD, expand (F ยฑ G)* LCD
and simplify (F ยฑ G)(LCD) / LCD.
48 28 + 30 โ 27
48
II. to simplify complex fractions
III. to solve rational equations (later)
85. Rational Expressions
The least common denominator (LCD) is needed
I. to combine (add or subtract) rational expressions
Example H: Combine
7
12
5
8
+ โ
16
9
The LCD = 48, ( )*
48
6
7
12
5
8
+ โ
16
9
4 3
=
Combining Rational Expressions (LCD Method):
To combine rational expressions (F ยฑ G),
multiple (F ยฑ G)* LCD/LCD, expand (F ยฑ G)* LCD
and simplify (F ยฑ G)(LCD) / LCD.
48 28 + 30 โ 27
48
=
48
31
II. to simplify complex fractions
III. to solve rational equations (later)
99. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
โ
4
1
3
2
100. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
, Their LCD is 12.
101. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
Their LCD is 12.
102. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
Their LCD is 12.
=
1
103. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
=
Their LCD is 12.
โ
3
1
2
3
โ
4
1
3
2
*12
*12
*12
*12
104. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
=
Their LCD is 12.
โ
3
1
2
3
โ
4
1
3
2
*12
*12 *12
*12 4
105. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
=
Their LCD is 12.
โ
3
1
2
3
โ
4
1
3
2
*12
*12 *12
*12 4 6
106. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
=
Their LCD is 12.
โ
3
1
2
3
โ
4
1
3
2
*12
*12 *12
*12 4 6
3 4
107. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
=
Their LCD is 12.
โ
3
1
2
3
โ
4
1
3
2
*12
*12 *12
*12 4 6
3 4
= 3
4 โ 18
โ 8
108. Rational Expressions
Example J. Simplify
โ
3
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
2
3
The fractional terms are
โ
4
1
3
2
3
1
2
3
4
1
3
2
.
,
,
,
Multiplying 12/12 (or 1) to the problem:
โ
3
1
2
3
โ
4
1
3
2
( )
)
(
12
12
=
Their LCD is 12.
โ
3
1
2
3
โ
4
1
3
2
*12
*12 *12
*12 4 6
3 4
= 3
4 โ 18
โ 8 = 14
5
109. Rational Expressions
Example K. Simplify
โ
(x โ h)
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
(x + h)
1
2h
Multiply the top and bottom by (x โ h)(x + h) to reduce the
expression in the numerators to polynomials.
โ
(x โ h)
1
(x + h)
1
2h
=
โ
(x โ h)
1
(x + h)
1
2h
(x + h)(x โ h)
[ ]
(x + h)(x โ h)
*
=
โ
(x + h) (x โ h)
2h(x + h)(x โ h)
=
2h
2h(x + h)(x โ h)
=
1
(x + h)(x โ h)
110. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
Rationalize Radicals
111. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
Rationalize Radicals
112. Example K: Rationalize the numerator
To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
h
๏x + h โ ๏x
Rationalize Radicals
113. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
Rationalize Radicals
h
๏x + h โ ๏x
= h
(๏x + h โ ๏x) (๏x + h + ๏x)
(๏x + h + ๏x)
*
Example K: Rationalize the numerator h
๏x + h โ ๏x
114. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
Rationalize Radicals
h
๏x + h โ ๏x
= h
(๏x + h โ ๏x) (๏x + h + ๏x)
(๏x + h + ๏x)
*
=
h
(๏x + h)2 โ (๏x)2
(๏x + h + ๏x)
Example K: Rationalize the numerator h
๏x + h โ ๏x
115. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
Rationalize Radicals
h
๏x + h โ ๏x
= h
(๏x + h โ ๏x) (๏x + h + ๏x)
(๏x + h + ๏x)
*
=
h
(๏x + h)2 โ (๏x)2
(๏x + h + ๏x)
Example K: Rationalize the numerator h
๏x + h โ ๏x
(x + h) โ (x) = h
116. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
Rationalize Radicals
h
๏x + h โ ๏x
= h
(๏x + h โ ๏x) (๏x + h + ๏x)
(๏x + h + ๏x)
*
=
h
(๏x + h)2 โ (๏x)2
(๏x + h + ๏x)
=
h
h
(๏x + h + ๏x)
Example K: Rationalize the numerator h
๏x + h โ ๏x
(x + h) โ (x) = h
117. To rationalize radicals in expressions we often use
the formula (x โ y)(x + y) = x2 โ y2.
(x + y) and (x โ y) are called conjugates.
Rationalize Radicals
h
๏x + h โ ๏x
= h
(๏x + h โ ๏x) (๏x + h + ๏x)
(๏x + h + ๏x)
*
=
h
(๏x + h)2 โ (๏x)2
(๏x + h + ๏x)
=
h
h
(๏x + h + ๏x)
=
1
๏x + h + ๏x
Example K: Rationalize the numerator h
๏x + h โ ๏x
(x + h) โ (x) = h
118. Exercise A. Factor each expression then use the factored
form to evaluate the given input values. No calculator.
Applications of Factoring
1. x2 โ 3x โ 4, x = โ2, 3, 5 2. x2 โ 2x โ 15, x = โ1, 4, 7
3. x2 โ x โ 2, x = ยฝ ,โ2, โยฝ 4. x3 โ 2x2, x = โ2, 2, 4
5. x4 โ 3x2, x = โ1, 1, 5 6. x3 โ 4x2 โ 5x, x = โ4, 2, 6
B. Determine if the output is positive or negative using the
factored form.
7.
x2 โ 4
x + 4
8. x3 โ 2x2
x2 โ 2x + 1
, x = โ3, 1, 5 , x = โ0.1, 1/2, 5
4.
x2 โ 4
x + 4 5. x2 + 2x โ 3
x2 + x
6. x3 โ 2x2
x2 โ 2x + 1
, x = โ3.1, 1.9 , x = โ0.1, 0.9, 1.05
, x = โ0.1, 0.99, 1.01
1. x2 โ 3x โ 4, x = โ2ยฝ, โ2/3, 2ยฝ, 5ยผ
2. โx2 + 2x + 8, x = โ2ยฝ, โ2/3, 2ยฝ, 5ยผ
3. x3 โ 2x2 โ 8x, x = โ4ยฝ, โ3/4, ยผ, 6ยผ,