The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial in a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial within a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
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.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are:
- Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x).
- Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (
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.
The document discusses applications of factoring expressions. The main purposes of factoring an expression E into a product E=AB is to utilize properties of multiplication. The most important application of factoring is to solve polynomial equations by setting each factor equal to 0 based on the zero-product property. Examples are provided to demonstrate solving polynomial equations by factoring, setting each factor equal to 0, and extracting the solutions.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial within a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
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.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are:
- Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x).
- Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (
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.
The document discusses applications of factoring expressions. The main purposes of factoring an expression E into a product E=AB is to utilize properties of multiplication. The most important application of factoring is to solve polynomial equations by setting each factor equal to 0 based on the zero-product property. Examples are provided to demonstrate solving polynomial equations by factoring, setting each factor equal to 0, and extracting the solutions.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses sign charts and how to determine the signs of outputs for polynomials and rational expressions. It provides examples of factoring polynomials to determine if the output is positive or negative for given values of x. The key steps to create a sign chart are: 1) solve for f=0 and any undefined values, 2) mark these values on a number line, 3) sample points in each segment to determine the sign in that region. Sign charts indicate the regions where a function is positive, negative or zero.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
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 formulas for multiplying binomial expressions. It states that the conjugate of expressions like (A + B) is (A - B). The difference of squares formula is given as (A + B)(A - B) = A^2 - B^2. Examples of expanding expressions using this formula and the square formulas (A + B)^2 = A^2 + 2AB + B^2 and (A - B)^2 = A^2 - 2AB + B^2 are provided.
The document defines an arithmetic sequence as a sequence where the nth term is defined by a linear formula of the form an = d*n + c. It provides examples of arithmetic sequences and explains the general formula for finding any term in an arithmetic sequence if the first term (a1) and the common difference (d) between terms are known. It demonstrates using the general formula to find the specific formula for various arithmetic sequences given parts of the sequence.
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 linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
47 operations of 2nd degree expressions and formulasalg1testreview
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.
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
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 rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They may be written in expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating expressions for given inputs, and determining the signs of outputs. The domain excludes values of x that make the denominator equal to 0.
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
1.0 factoring trinomials the ac method and making lists-xmath260
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.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
This document provides information about quadratic equations, including:
- Methods for solving quadratic equations like factoring, completing the square, and using the quadratic formula.
- Key terms like discriminant and nature of roots. The discriminant determines if the roots are real, equal, or imaginary.
- Examples of solving quadratic equations using different methods and finding related values like discriminant and roots.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
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 methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document discusses using the factor theorem to factor polynomials. It provides examples of finding the factors of a polynomial given its zeros. It also presents the factor-solution-intercept equivalence theorem, which states that for any polynomial f, the following are equivalent: (x - c) is a factor of f, f(c) = 0, c is an x-intercept of the graph y = f(x), c is a zero of f, and the remainder when f(x) is divided by (x - c) is 0. Examples are worked through to demonstrate factoring polynomials by finding zeros and dividing.
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.
The document provides an overview of key concepts in algebra including:
1) Real numbers which are made up of rational and irrational numbers. Real numbers have properties like closure under addition and multiplication.
2) Exponents and radicals including laws involving integral, zero, fractional exponents and radicals.
3) Polynomials which are algebraic expressions made up of variables and coefficients, and can be added or subtracted by combining like terms.
This document contains information about polynomials:
- A polynomial is an expression with variables and constants using only addition, subtraction, multiplication, and non-negative integer exponents. Each term is a constant multiplied by one or more variables raised to a non-negative integer power.
- The degree of a polynomial is equal to the highest exponent in any term. Terms are usually written in descending order of degree.
- If two polynomials are equal for all values of the variable, then the coefficients of like powers must be equal, making the polynomials identical.
The document discusses sign charts and how to determine the signs of outputs for polynomials and rational expressions. It provides examples of factoring polynomials to determine if the output is positive or negative for given values of x. The key steps to create a sign chart are: 1) solve for f=0 and any undefined values, 2) mark these values on a number line, 3) sample points in each segment to determine the sign in that region. Sign charts indicate the regions where a function is positive, negative or zero.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
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 formulas for multiplying binomial expressions. It states that the conjugate of expressions like (A + B) is (A - B). The difference of squares formula is given as (A + B)(A - B) = A^2 - B^2. Examples of expanding expressions using this formula and the square formulas (A + B)^2 = A^2 + 2AB + B^2 and (A - B)^2 = A^2 - 2AB + B^2 are provided.
The document defines an arithmetic sequence as a sequence where the nth term is defined by a linear formula of the form an = d*n + c. It provides examples of arithmetic sequences and explains the general formula for finding any term in an arithmetic sequence if the first term (a1) and the common difference (d) between terms are known. It demonstrates using the general formula to find the specific formula for various arithmetic sequences given parts of the sequence.
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 linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
47 operations of 2nd degree expressions and formulasalg1testreview
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.
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
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 rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They may be written in expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating expressions for given inputs, and determining the signs of outputs. The domain excludes values of x that make the denominator equal to 0.
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
1.0 factoring trinomials the ac method and making lists-xmath260
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.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
This document provides information about quadratic equations, including:
- Methods for solving quadratic equations like factoring, completing the square, and using the quadratic formula.
- Key terms like discriminant and nature of roots. The discriminant determines if the roots are real, equal, or imaginary.
- Examples of solving quadratic equations using different methods and finding related values like discriminant and roots.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
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 methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document discusses using the factor theorem to factor polynomials. It provides examples of finding the factors of a polynomial given its zeros. It also presents the factor-solution-intercept equivalence theorem, which states that for any polynomial f, the following are equivalent: (x - c) is a factor of f, f(c) = 0, c is an x-intercept of the graph y = f(x), c is a zero of f, and the remainder when f(x) is divided by (x - c) is 0. Examples are worked through to demonstrate factoring polynomials by finding zeros and dividing.
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.
The document provides an overview of key concepts in algebra including:
1) Real numbers which are made up of rational and irrational numbers. Real numbers have properties like closure under addition and multiplication.
2) Exponents and radicals including laws involving integral, zero, fractional exponents and radicals.
3) Polynomials which are algebraic expressions made up of variables and coefficients, and can be added or subtracted by combining like terms.
This document contains information about polynomials:
- A polynomial is an expression with variables and constants using only addition, subtraction, multiplication, and non-negative integer exponents. Each term is a constant multiplied by one or more variables raised to a non-negative integer power.
- The degree of a polynomial is equal to the highest exponent in any term. Terms are usually written in descending order of degree.
- If two polynomials are equal for all values of the variable, then the coefficients of like powers must be equal, making the polynomials identical.
Algebra is the study of mathematical symbols and rules for calculating those symbols, which allows numbers to be represented by variables. An algebraic expression combines constants and variables using operations like addition, subtraction, multiplication and division. Expressions can be monomials with one term, binomials with two terms, or trinomials with three terms. To multiply algebraic expressions, the signs and coefficients are multiplied, and the variables are multiplied using exponent rules.
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.
The document discusses several topics in algebra including:
1. Indices laws including am x an = am + n, am ÷ an = am - n, and (am)n = amn. Negative and fractional indices are also discussed.
2. Logarithms including the definition that logarithm of 'x' to base 'a' is the power to which 'a' must be raised to give 'x'. Change of base formula is also provided.
3. Series including the definition of finite and infinite series. Notation of sigma notation ∑ is introduced to represent the sum of terms.
This document provides information about polynomials including definitions, types, terms, and relationships between coefficients and zeros. It begins with acknowledging those who helped create the presentation. It then defines a polynomial as an expression with variable terms raised to whole number powers. The main types discussed are linear, quadratic, and cubic polynomials. Linear polynomials have one zero while quadratics have two zeros and cubics have three. Relationships are defined between the zeros and coefficients. Graphs of linear and quadratic polynomials are presented. The division algorithm for polynomials is also explained.
The document discusses various topics in advanced algebra including inequalities, arithmetic progressions, geometric progressions, harmonic progressions, permutations, combinations, matrices, determinants, and solving systems of linear equations using matrices. Key properties and formulas are provided for each topic. Examples are included to demonstrate solving problems related to each concept.
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.
Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.
The document provides information about polynomials including:
1) A polynomial is an expression constructed from variables and constants using operations of addition, subtraction, multiplication, and exponents.
2) The degree of a polynomial refers to the exponent of its highest term. For example, a quadratic polynomial is degree 2 and a cubic is degree 3.
3) The zeros of a polynomial are the values that make the polynomial equal to 0. Finding the zeros involves solving the polynomial equation.
4) The relationships between the zeros and coefficients of a polynomial can be used to find unknown coefficients or zeros.
factoring trinomials the ac method and making listsmath260
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.
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
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.
Polynomials are mathematical expressions constructed from variables and constants using addition, subtraction, multiplication, and exponents of whole numbers. They appear in many areas of mathematics and science. Polynomials can be used to form equations that model problems in various domains. They also define polynomial functions that are used in fields like physics, chemistry, economics, and social sciences. Polynomials are classified based on their degree, with linear polynomials having degree 1 and quadratic polynomials degree 2. The maximum number of zeroes a polynomial can have is equal to its degree.
Are you scared of that algebraic sums? Just view this presentation an you can learn about each and every algebraic identities. Just view this and Now take full marks in your tests..
This document provides an overview of real numbers and exponents in Algebra 1. It defines rational and irrational numbers, and gives examples of each. Rules for exponents and radicals are presented, including laws for integral, fractional, zero exponents and properties of radicals like addition, subtraction, multiplication and division. Examples are provided to demonstrate each rule or property. Practice problems are included for students to simplify expressions using exponent and radical rules.
- Polynomials are expressions constructed from variables and constants with non-negative whole number exponents.
- The degree of a polynomial is the highest exponent among its terms. Zeroes are values that make the polynomial equal to zero.
- There is a relationship between the number of zeroes a polynomial can have and its degree. Linear polynomials have at most 1 zero, quadratics have at most 2 zeros, and cubics have at most 3 zeros.
- The coefficients of a polynomial are related to its zeroes through formulas involving the sum and product of the zeroes.
This document provides an overview of algebraic expressions and operations with polynomials. It defines algebraic expressions as combinations of letters and numbers using operations like addition, subtraction, multiplication, and exponents. Monomials are single-term expressions while polynomials can have multiple terms. The key operations discussed are adding and multiplying polynomials by distributing terms. Factoring polynomials uses techniques like removing common factors, applying identities like the difference of squares, and recognizing perfect square trinomials.
This document provides an overview of algebraic expressions and operations with algebraic expressions including:
- Algebraic expressions involve unknown quantities represented by letters combined with numbers and operations.
- The numerical value of an expression is found by substituting values for the unknowns.
- Types of expressions include monomials, binomials, and polynomials.
- Common operations with expressions include addition, multiplication, division, and taking powers of expressions.
- Polynomials can be added, multiplied, and factors can be removed through operations like difference of squares identities.
Similar to 6 polynomial expressions and operations (20)
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
The document discusses the concept of slope of a line. It defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically:
- The slope of a line is calculated as the change in the y-values (rise) divided by the change in the x-values (run) between two points on the line.
- This formula is easy to memorize and captures the geometric meaning of slope as the tilt of the line.
- An example problem demonstrates calculating the slope of a line between two points by finding the difference in their x- and y-values.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
2 the real line, inequalities and comparative phraseselem-alg-sample
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Examples are provided of drawing intervals on the number line and solving simple inequalities algebraically. Properties of inequalities like adding the same quantity to both sides preserving the inequality sign are also outlined.
Geometry is the study of shapes, their properties and relationships. Some basic geometric shapes include lines, rays, angles, triangles, quadrilaterals, polygons, circles and three-dimensional shapes like spheres and cubes. Formulas are used to calculate properties of shapes like the area of a triangle is 1/2 * base * height, the circumference of a circle is 2 * pi * radius, and the volume of a cube is side^3.
The document discusses direct and inverse variations. It defines a direct variation as a relationship where y=kx, where k is a constant. An inverse variation is defined as a relationship where y=k/x, where k is a constant. Examples are given of translating phrases describing direct and inverse variations into mathematical equations. The document also explains how to solve word problems involving variations by using given values to find the specific constant k and exact variation equation.
17 applications of proportions and the rational equationselem-alg-sample
The document discusses rational equations word problems involving rates, distances, costs, and number of people. An example problem asks how many people (x) shared a taxi costing $20 if one person leaving causes the remaining people's cost to increase by $1 each. Setting up rational equations and solving leads to the answer that x = 5 people.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
15 proportions and the multiplier method for solving rational equationselem-alg-sample
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding or subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate converting rational expressions to equivalent forms with different specified denominators.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
Trinomials are polynomials of the form ax^2 + bx + c, where a, b, and c are numbers. To factor a trinomial, we write it as the product of two binomials (x + u)(x + v) where uv = c and u + v = b. For example, to factor x^2 + 5x + 6, we set uv = 6 and u + v = 5. The only possible values are u = 2 and v = 3, so x^2 + 5x + 6 = (x + 2)(x + 3). Similarly, to factor x^2 - 5x + 6, we set uv = 6 and u + v = -5,
The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.
The document discusses methods for multiplying binomial expressions. A binomial is a two-term polynomial of the form ax + b, while a trinomial is a three-term polynomial of the form ax^2 + bx + c. The product of two binomials results in a trinomial. The FOIL method is introduced to multiply binomials, where the Front, Outer, Inner, and Last terms of each binomial are multiplied and combined. Expanding the product of a binomial and a binomial with a leading negative sign requires distributing the negative sign first before using FOIL.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then presents and explains the following rules for exponents:
1) Multiplication Rule: ANAK = AN+K
2) Division Rule: AN/AK = AN-K
3) Power Rule: (AN)K = ANK
4) 0-Power Rule: A0 = 1
5) Negative Power Rule: A-K = 1/AK
It provides examples to illustrate how to apply each rule when simplifying expressions with exponents.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides steps to take which include clearing fractions by multiplying both sides by the LCD, moving all terms except the variable of interest to one side of the equation, and then dividing both sides by the coefficient of the isolated variable term to solve for the variable. Examples are provided to demonstrate these steps, such as solving for x in (a + b)x = c by dividing both sides by (a + b).
The document provides examples and explanations for solving linear equations with one step. It defines a linear equation as one where both sides are linear expressions, such as 3x + 10 = 34, and not containing higher powers of x. To solve a one-step linear equation, the goal is to isolate the variable x on one side by applying the opposite operation to both sides, such as adding 3 to both sides of x - 3 = 12 to get x = 15. Worked examples are provided for solving equations of the form x ± a = b and cx = d.
2. A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
3. Example A.
2 + 3x
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
4. Example A.
2 + 3x “the sum of 2 and 3 times x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
5. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
6. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
7. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
8. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Polynomial Expressions
9. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
Polynomial Expressions
10. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
11. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4
a. 3y2
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
12. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4
a. 3y2
3y2 3(–4)2
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
13. Example A.
2 + 3x “the sum of 2 and 3 times x”
4x2 – 5x “the difference between 4 times the square of x
and 5 times x”
(3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in
numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4
a. 3y2
3y2 3(–4)2
= 3(16) = 48
An expression of the form #xN, where the exponent N is a
non-negative integer and # is a number, is called a monomial
(one-term).
For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
22. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
Polynomial Expressions
Polynomial Expressions
23. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7,
Polynomial Expressions
Polynomial Expressions
24. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7,
Polynomial Expressions
Polynomial Expressions
25. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,
Polynomial Expressions
Polynomial Expressions
26. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms),
these are expressions of the form
#xN ± #xN-1 ± … ± #x1 ± #
where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,
x
1
is not a polynomial.whereas the expression
Polynomial Expressions
Polynomial Expressions
27. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
Polynomial Expressions
28. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3.
Polynomial Expressions
29. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Polynomial Expressions
30. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression,
Polynomial Expressions
31. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
Polynomial Expressions
32. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
Polynomial Expressions
33. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Polynomial Expressions
34. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
Polynomial Expressions
35. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Polynomial Expressions
36. Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Therefore the polynomial –3x2 – 4x + 7 has 3 terms,
–3x2 , –4x and + 7.
Polynomial Expressions
37. Each term is addressed by the variable part.
Polynomial Expressions
38. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2,
Polynomial Expressions
39. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
Polynomial Expressions
40. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
Polynomial Expressions
41. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term.
Polynomial Expressions
42. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Polynomial Expressions
43. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Operations with Polynomials
Polynomial Expressions
44. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Operations with Polynomials
Polynomial Expressions
45. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
Operations with Polynomials
Polynomial Expressions
46. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x
Operations with Polynomials
Polynomial Expressions
47. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Operations with Polynomials
Polynomial Expressions
48. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined.
Operations with Polynomials
Polynomial Expressions
49. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Operations with Polynomials
Polynomial Expressions
50. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
Operations with Polynomials
Polynomial Expressions
51. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient.
Operations with Polynomials
Polynomial Expressions
52. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x
Operations with Polynomials
Polynomial Expressions
53. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
Operations with Polynomials
Polynomial Expressions
54. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
Polynomial Expressions
55. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
When multiplying a number with a polynomial, we may
expand using the distributive law: A(B ± C) = AB ± AC.
Polynomial Expressions
56. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
Polynomial Expressions
57. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
Polynomial Expressions
58. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
Polynomial Expressions
59. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
Polynomial Expressions
62. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
63. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) =
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
64. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
65. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
66. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x)
=
67. Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x) distribute
= 6x5 – 12x3
68. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
69. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
a. (3x + 2)(2x – 1)
70. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
= 3x(2x – 1) + 2(2x – 1)
a. (3x + 2)(2x – 1)
71. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
a. (3x + 2)(2x – 1)
72. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
a. (3x + 2)(2x – 1)
73. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
a. (3x + 2)(2x – 1)
74. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
a. (3x + 2)(2x – 1)
75. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
a. (3x + 2)(2x – 1)
76. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
77. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
78. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ ≡ QP.
79. To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ ≡ QP.
A shorter way to multiply is to bypass the 2nd step and use the
general distributive law.
84. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Polynomial Operations
85. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
Polynomial Operations
86. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2
Polynomial Operations
87. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x
Polynomial Operations
88. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x
Polynomial Operations
89. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12
Polynomial Operations
90. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
Polynomial Operations
91. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
92. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3
93. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2
94. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x
95. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2
96. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x
97. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6
98. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6
= x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-
after we understand more about the multiplication operation.
99. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6
= x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-
after we understand more about the multiplication operation.
101. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Polynomial Expressions
102. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
103. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
104. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
105. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
106. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
107. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
108. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
109. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
110. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2 = –16y – 6y2
111. Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
If both numbers are given, then we get a numerical output.
We may do this for x, y and z or even more variables.
Polynomial Expressions
Example H. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2 = –16y – 6y2
112. Ex. A. Evaluate each monomials with the given values.
3. 2x2 with x = 1 and x = –1 4. –2x2 with x = 1 and x = –1
5. 5y3 with y = 2 and y = –2 6. –5y3 with y = 2 and y = –2
1. 2x with x = 1 and x = –1 2. –2x with x = 1 and x = –1
7. 5z4 with z = 2 and z = –2 8. –5y4 with z = 2 and z = –2
B. Evaluate each monomials with the given values.
9. 2x2 – 3x + 2 with x = 1 and x = –1
10. –2x2 + 4x – 1 with x = 2 and x = –2
11. 3x2 – x – 2 with x = 3 and x = –3
12. –3x2 – x + 2 with x = 3 and x = –3
13. –2x3 – x2 + 4 with x = 2 and x = –2
14. –2x3 – 5x2 – 5 with x = 3 and x = –3
C. Expand and simplify.
15. 5(2x – 4) + 3(4 – 5x) 16. 5(2x – 4) – 3(4 – 5x)
17. –2(3x – 8) + 3(4 – 9x) 18. –2(3x – 8) – 3(4 – 9x)
19. 7(–2x – 7) – 3(4 – 3x) 20. –5(–2 – 8x) + 7(–2 – 11x)
Polynomial Expressions
113. 21. x2 – 3x + 5 + 2(–x2 – 4x – 6)
22. x2 – 3x + 5 – 2(–x2 – 4x – 6)
23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6)
24. 2(x2 – 3x + 5) – 5(–x2 – 4x – 6)
25. –2(3x2 – 2x + 5) + 5(–4x2 – 4x – 3)
26. –2(3x2 – 2x + 5) – 5(–4x2 – 4x – 3)
27. 4(3x3 – 5x2) – 9(6x2 – 7x) – 5(– 8x – 2)
29. Simplify 2(3xy – xy2) – 2(2xy – xy2) then evaluated it
with x = –1, afterwards evaluate it at (–1, 2) for (x, y)
30. Simplify x2 – 2(3xy – x2) – 2(y2 – xy) then evaluated it
with y = –2, afterwards evaluate it at (–1, –2) for (x, y)
31. Simplify x2 – 2(3xy – z2) – 2(z2 – x2) then evaluated it
with x = –1, y = – 2 and z = 3.
Polynomial Expressions
28. –6(7x2 + 5x – 9) – 7(–3x2 – 2x – 7)