8.28 simplifying presentation
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This document provides examples for simplifying algebraic expressions by combining like terms. The first example expression is 8x - 20y + 12xy, which can be simplified to 8x + 20y + 12xy. The second example asks to simplify the expression 3x + 6y + 10xy + 3y, which combines to 3x + 10y + 10xy. Simplifying expressions involves combining terms with the same variables and exponents.
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March 6
The document provides a summary of factoring methods:
- Reviewing factoring methods covered such as greatest common factor (GCF) and factoring trinomials
- Announcing that test grades have been posted for one class and quarter grades will be posted for another class tomorrow
- Introducing a new factoring method called "difference of squares" and new Khan Academy topics being added
Working with polynomials
- This document discusses working with polynomials, including defining terms, writing polynomials, simplifying polynomials, adding and subtracting polynomials, multiplying polynomials using the FOIL method, and providing examples of each.
- Key aspects of polynomials include terms consisting of a variable and its exponent and coefficient, writing polynomials in order of decreasing power with the leading and constant terms defined, and combining like terms when simplifying, adding, or subtracting polynomials.
- To multiply polynomials, the FOIL method is used, which involves multiplying the First, Outer, Inner, and Last terms of binomial expressions.
Adding and subtracting polynomials
Adding and subtracting polynomials involves combining like terms. Like terms are terms that have the same variables and the same exponents. To add polynomials, terms are grouped by their like terms and the coefficients are combined by adding them. To subtract polynomials, the process is the same as adding the opposite of the second polynomial. The opposite of each term is found by changing its sign. Then the like terms are combined in the same way as when adding.
Polynomials
This document discusses polynomials and their properties. It defines polynomials as algebraic expressions involving terms of varying degrees, and defines the degree of a polynomial. It then covers operations like addition, multiplication, and factoring of polynomials using various properties and methods like FOIL. Examples are provided to demonstrate multiplying binomials and special products involving sums and differences of terms.
3/1/12 Factor by Grouping and Factoring into Quadratic Form
The document discusses factoring polynomials by grouping and factoring polynomials into quadratic form. It provides examples of factoring polynomials using these methods, such as factoring x3 - 3x2 - 16x + 48 as (x - 3)(x2 - 16). Students are asked to complete factoring exercises and solve a challenge problem involving factoring the volume equation of a jewelry box to find its dimensions.
125 7.7
This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
Combining like terms
The document defines exponents and provides examples of multiplying and combining like terms with exponents. It explains that to multiply terms, you multiply the coefficients and add the exponents of like terms. It provides examples of multiplying, combining, and dividing like terms with variables and exponents.
March 17, 2015
Today's class will include a warm-up, factoring polynomials using the greatest common factor (GCF) method and factoring by grouping, and Khan Academy assignments due tonight. Students should show all their work to receive credit for class work involving factoring polynomials using GCF and grouping methods.
Recommended
March 6
The document provides a summary of factoring methods:
- Reviewing factoring methods covered such as greatest common factor (GCF) and factoring trinomials
- Announcing that test grades have been posted for one class and quarter grades will be posted for another class tomorrow
- Introducing a new factoring method called "difference of squares" and new Khan Academy topics being added
Working with polynomials
- This document discusses working with polynomials, including defining terms, writing polynomials, simplifying polynomials, adding and subtracting polynomials, multiplying polynomials using the FOIL method, and providing examples of each.
- Key aspects of polynomials include terms consisting of a variable and its exponent and coefficient, writing polynomials in order of decreasing power with the leading and constant terms defined, and combining like terms when simplifying, adding, or subtracting polynomials.
- To multiply polynomials, the FOIL method is used, which involves multiplying the First, Outer, Inner, and Last terms of binomial expressions.
Adding and subtracting polynomials
Adding and subtracting polynomials involves combining like terms. Like terms are terms that have the same variables and the same exponents. To add polynomials, terms are grouped by their like terms and the coefficients are combined by adding them. To subtract polynomials, the process is the same as adding the opposite of the second polynomial. The opposite of each term is found by changing its sign. Then the like terms are combined in the same way as when adding.
Polynomials
This document discusses polynomials and their properties. It defines polynomials as algebraic expressions involving terms of varying degrees, and defines the degree of a polynomial. It then covers operations like addition, multiplication, and factoring of polynomials using various properties and methods like FOIL. Examples are provided to demonstrate multiplying binomials and special products involving sums and differences of terms.
3/1/12 Factor by Grouping and Factoring into Quadratic Form
The document discusses factoring polynomials by grouping and factoring polynomials into quadratic form. It provides examples of factoring polynomials using these methods, such as factoring x3 - 3x2 - 16x + 48 as (x - 3)(x2 - 16). Students are asked to complete factoring exercises and solve a challenge problem involving factoring the volume equation of a jewelry box to find its dimensions.
125 7.7
This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
Combining like terms
The document defines exponents and provides examples of multiplying and combining like terms with exponents. It explains that to multiply terms, you multiply the coefficients and add the exponents of like terms. It provides examples of multiplying, combining, and dividing like terms with variables and exponents.
March 17, 2015
Today's class will include a warm-up, factoring polynomials using the greatest common factor (GCF) method and factoring by grouping, and Khan Academy assignments due tonight. Students should show all their work to receive credit for class work involving factoring polynomials using GCF and grouping methods.
Adding and subtracting polynomials
This document discusses adding and subtracting polynomials. It explains that to add polynomials, terms with the same variables and exponents are combined. To subtract polynomials, the opposite of the second polynomial is taken and then the polynomials are added like normal. Examples are provided of rearranging terms in decreasing exponential order, stacking like terms, and combining them to solve polynomial addition and subtraction problems.
March 18 Radicals
This document discusses square roots and their properties. It provides examples of taking the square root of both sides of an equation to solve for the variable. It also shows combining like terms within square root expressions and then taking the square root of both sides to isolate the variable.
Sum and difference of two squares
(1) The sum and difference of two terms can be expressed as the square of the first term minus the square of the second term, as shown in the formula (x + y)(x - y) = x^2 - y^2.
(2) Several examples are provided to illustrate multiplying the sum and difference of terms, which results in the difference of the squares of the two terms.
(3) Multiplying expressions like (x - 2)(x + 2), (2x + 5)(2x - 5), (3x + y)(3x - y), and (xy4 - 3)(xy4 + 3) follows the pattern of being equal to the difference of
Unit 4 lesson 8 dist property combining like terms
This document provides 18 math word problems involving distributing terms and combining like terms. Students are asked to simplify expressions by distributing terms within parentheses and combining like terms, then evaluate the expressions for given variable values. The problems cover a variety of expressions including those with addition, subtraction, multiplication, and multiple sets of parentheses.
Foil method
The document provides examples of using the FOIL (First, Outer, Inner, Last) method to multiply binomial expressions. It shows step-by-step work for multiplying (x + 2)(x - 4) and (x - 3)(x - 2) using FOIL. It then lists three additional practice problems for multiplying binomial expressions using FOIL without showing the steps.
Foil method
The document provides examples of using the FOIL (First, Outer, Inner, Last) method to multiply binomial expressions. It shows step-by-step work for multiplying (x + 2)(x - 4) and (x - 3)(x - 2) using FOIL. It then lists three additional practice problems for multiplying binomial expressions using FOIL without showing the steps.
Lesson 6 subtraction of polynomials
The document summarizes how to subtract polynomials. It explains that to subtract polynomials, one changes the sign of all terms in the subtrahend, changes the operation to addition, and then proceeds to add the polynomials by aligning like terms and combining coefficients. It provides examples of subtracting various polynomials, as well as exercises for students to practice subtracting polynomials.
Product of a binomial and a trinomial involving
The document discusses how to find the sum or difference of two cubes by multiplying a binomial with a trinomial of a certain form. It provides examples of multiplying binomial expressions of the form (a - b) with trinomials of the form (a^2 + ab + b^2) and using the rule (a - b)(a^2 + ab + b^2) = (a^3 - b^3). It also gives an example that does not follow this rule.
Exit slip
This document contains 4 math equations with variables on both sides of the equal sign. Each equation has variables x along with addition or subtraction of other terms, such as numbers or additional variables, on both sides of the equal sign. The goal appears to be solving each equation for the variable x.
Square of a binomial
The document discusses how to find the square of a binomial expression using the formula F2 + 2FL + L2, where F is the first term, L is the last term, and examples are provided to demonstrate its use. The key steps are: (1) square the first term for F2, (2) multiply the first and last terms and double the product for 2FL, and (3) square the last term for L2. Several examples of finding the square of binomial expressions such as (x + 2)2, (x - 3)2, and (3 - 6x)2 are worked through to illustrate the application of the formula.
P6 factoring
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.
Ankit maths ppt
A polynomial is an algebraic expression involving variables and constants that is formed using addition, subtraction, and multiplication, but not division. The degree of a polynomial refers to the highest power of the variable. Polynomials can be classified based on their number of terms as monomials (one term), binomials (two terms), or trinomials (three terms). They can also be classified based on degree as constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3), or of higher degrees. The number of zeros of a polynomial is equal to its degree.
Factoring general trinomials
This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.
Cube of binomial
The cube of a binomial can be found using the formula F3 + 3F2L + 3FL2 + L3, where:
- F3 is the cube of the first term
- 3F2L is 3 times the square of the first term multiplied by the second term
- 3FL2 is 3 times the first term multiplied by the square of the second term
- L3 is the cube of the second term
This formula is demonstrated through examples of finding the cubes of (x + 2), (x - 2), and (2x + y).
Adding and subtracting polynomials
This document provides an overview of adding and subtracting polynomials. It defines monomials, binomials, and trinomials as polynomials with 1, 2, or 3 terms respectively. Polynomials can have any number of terms. Terms must have like variables to be combined using addition or subtraction. Examples are provided for determining the type of polynomial, simplifying polynomials by combining like terms, and adding and subtracting polynomials. Practice problems are assigned from the workbook.
Common factor factorization
This presentation discusses factoring polynomials by finding the greatest common factor. It defines key terms like factors, common factors, and greatest common factor. It explains how to find all the factors of a number and determine common factors between two numbers. Examples are provided of factoring polynomials by identifying the greatest common monomial factor and factoring it out of each term. Steps for factoring a monomial out of a polynomial are outlined.
Factor by grouping
The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.
Add,sub,mult polynomials
1) To add or subtract polynomials, combine like terms by adding or subtracting their coefficients.
2) To multiply polynomials, use FOIL or the distributive property to multiply each term of one factor with each term of the other.
3) Important patterns for multiplying binomials include the sum and difference pattern, square of a binomial, and cube of a binomial.
Higher order polynomials_ppt_2
This document discusses higher order polynomials through two examples. The first example provides the cubic equation f(x)=x^3-3x^2+4x-12, finds its roots through factoring, and graphs the equation. The second example gives the quartic equation f(x)=x^4+3x^2+4, finds its four roots including two complex roots through factoring, and graphs the equation.
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Adding and subtracting polynomials
This document discusses adding and subtracting polynomials. It explains that to add polynomials, terms with the same variables and exponents are combined. To subtract polynomials, the opposite of the second polynomial is taken and then the polynomials are added like normal. Examples are provided of rearranging terms in decreasing exponential order, stacking like terms, and combining them to solve polynomial addition and subtraction problems.
March 18 Radicals
This document discusses square roots and their properties. It provides examples of taking the square root of both sides of an equation to solve for the variable. It also shows combining like terms within square root expressions and then taking the square root of both sides to isolate the variable.
Sum and difference of two squares
(1) The sum and difference of two terms can be expressed as the square of the first term minus the square of the second term, as shown in the formula (x + y)(x - y) = x^2 - y^2.
(2) Several examples are provided to illustrate multiplying the sum and difference of terms, which results in the difference of the squares of the two terms.
(3) Multiplying expressions like (x - 2)(x + 2), (2x + 5)(2x - 5), (3x + y)(3x - y), and (xy4 - 3)(xy4 + 3) follows the pattern of being equal to the difference of
Unit 4 lesson 8 dist property combining like terms
This document provides 18 math word problems involving distributing terms and combining like terms. Students are asked to simplify expressions by distributing terms within parentheses and combining like terms, then evaluate the expressions for given variable values. The problems cover a variety of expressions including those with addition, subtraction, multiplication, and multiple sets of parentheses.
Foil method
The document provides examples of using the FOIL (First, Outer, Inner, Last) method to multiply binomial expressions. It shows step-by-step work for multiplying (x + 2)(x - 4) and (x - 3)(x - 2) using FOIL. It then lists three additional practice problems for multiplying binomial expressions using FOIL without showing the steps.
Foil method
The document provides examples of using the FOIL (First, Outer, Inner, Last) method to multiply binomial expressions. It shows step-by-step work for multiplying (x + 2)(x - 4) and (x - 3)(x - 2) using FOIL. It then lists three additional practice problems for multiplying binomial expressions using FOIL without showing the steps.
Lesson 6 subtraction of polynomials
The document summarizes how to subtract polynomials. It explains that to subtract polynomials, one changes the sign of all terms in the subtrahend, changes the operation to addition, and then proceeds to add the polynomials by aligning like terms and combining coefficients. It provides examples of subtracting various polynomials, as well as exercises for students to practice subtracting polynomials.
Product of a binomial and a trinomial involving
The document discusses how to find the sum or difference of two cubes by multiplying a binomial with a trinomial of a certain form. It provides examples of multiplying binomial expressions of the form (a - b) with trinomials of the form (a^2 + ab + b^2) and using the rule (a - b)(a^2 + ab + b^2) = (a^3 - b^3). It also gives an example that does not follow this rule.
Exit slip
This document contains 4 math equations with variables on both sides of the equal sign. Each equation has variables x along with addition or subtraction of other terms, such as numbers or additional variables, on both sides of the equal sign. The goal appears to be solving each equation for the variable x.
Square of a binomial
The document discusses how to find the square of a binomial expression using the formula F2 + 2FL + L2, where F is the first term, L is the last term, and examples are provided to demonstrate its use. The key steps are: (1) square the first term for F2, (2) multiply the first and last terms and double the product for 2FL, and (3) square the last term for L2. Several examples of finding the square of binomial expressions such as (x + 2)2, (x - 3)2, and (3 - 6x)2 are worked through to illustrate the application of the formula.
P6 factoring
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.
Ankit maths ppt
A polynomial is an algebraic expression involving variables and constants that is formed using addition, subtraction, and multiplication, but not division. The degree of a polynomial refers to the highest power of the variable. Polynomials can be classified based on their number of terms as monomials (one term), binomials (two terms), or trinomials (three terms). They can also be classified based on degree as constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3), or of higher degrees. The number of zeros of a polynomial is equal to its degree.
Factoring general trinomials
This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.
Cube of binomial
The cube of a binomial can be found using the formula F3 + 3F2L + 3FL2 + L3, where:
- F3 is the cube of the first term
- 3F2L is 3 times the square of the first term multiplied by the second term
- 3FL2 is 3 times the first term multiplied by the square of the second term
- L3 is the cube of the second term
This formula is demonstrated through examples of finding the cubes of (x + 2), (x - 2), and (2x + y).
Adding and subtracting polynomials
This document provides an overview of adding and subtracting polynomials. It defines monomials, binomials, and trinomials as polynomials with 1, 2, or 3 terms respectively. Polynomials can have any number of terms. Terms must have like variables to be combined using addition or subtraction. Examples are provided for determining the type of polynomial, simplifying polynomials by combining like terms, and adding and subtracting polynomials. Practice problems are assigned from the workbook.
Common factor factorization
This presentation discusses factoring polynomials by finding the greatest common factor. It defines key terms like factors, common factors, and greatest common factor. It explains how to find all the factors of a number and determine common factors between two numbers. Examples are provided of factoring polynomials by identifying the greatest common monomial factor and factoring it out of each term. Steps for factoring a monomial out of a polynomial are outlined.
Factor by grouping
The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.
Add,sub,mult polynomials
1) To add or subtract polynomials, combine like terms by adding or subtracting their coefficients.
2) To multiply polynomials, use FOIL or the distributive property to multiply each term of one factor with each term of the other.
3) Important patterns for multiplying binomials include the sum and difference pattern, square of a binomial, and cube of a binomial.
Higher order polynomials_ppt_2
This document discusses higher order polynomials through two examples. The first example provides the cubic equation f(x)=x^3-3x^2+4x-12, finds its roots through factoring, and graphs the equation. The second example gives the quartic equation f(x)=x^4+3x^2+4, finds its four roots including two complex roots through factoring, and graphs the equation.
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Unit 4 lesson 8 dist property combining like terms
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8.28 simplifying presentation
- 2. Example 1 8x 20y 12xy Simplify the expression. 4(2x + 5y + 3xy) A. 8x + 20y +12xy B. 8x +10y +12xy C. 6x +10y -12xy D. 6x + 9y -12xy ____ ____ ______
- 3. Example 2
- 4. Example 3 Combine like terms 3x + 6y + 10xy + 3y A. 9x + 10xy + 3y B. 3x + 9y + 10xy C. 3x +10y + 10xy D. 3x + 18y + 10xy