Factoring polynomials
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This document discusses factoring polynomials. It explains that factoring is reversing the process of multiplying monomials, binomials, or trinomials. The document focuses on factoring out the greatest common factor (GCF) from a polynomial and factoring trinomials into two binomials. It provides examples of factoring polynomials by finding the GCF, factoring the first and last terms of a trinomial, and checking that the signs are correct when factoring. It emphasizes that the signs must be placed correctly when factoring polynomials.
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factoring polynomials
This document outlines key concepts and examples for factoring polynomials. It covers factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL multiplication. Sections include the greatest common factor, factoring trinomials of the forms x^2 + bx + c and ax^2 + bx + c.
Factoring Polynomials to find its zeros
This document provides a lecture on factoring polynomials to find zeros. It begins with the lecture objectives of learning how to find the greatest common factor of a polynomial and factor simple quadratics to reveal their zeros. It then discusses finding the greatest common factor, factoring using the difference of squares formula, and factoring simple quadratics by finding two numbers that multiply and add to given values. Examples are provided for each technique. The document concludes by having students practice these skills by factoring and finding the zeros of polynomials.
Factoring polynomials
This document discusses factoring polynomials. It explains that factoring is reversing the process of multiplication to express a polynomial as a product of binomials. It provides examples of factoring out the greatest common factor (GCF) from a polynomial and factoring trinomials into two binomials. It outlines the step-by-step process of factoring trinomials, emphasizing the importance of checking signs. Finally, it provides examples for the reader to practice factoring polynomials.
Polynomials and factoring
The document discusses polynomials and factoring polynomials. It defines polynomials as expressions with terms added or subtracted, where terms are products of numbers and variables with exponents. It provides examples of monomials, binomials, trinomials, and polynomials based on the number of terms. It also discusses finding the greatest common factor of a polynomial to factor out a monomial.
Factoring quadratic trinomial
The document discusses factoring quadratic trinomials. It provides three examples of factoring expressions such as x^2 - 5x - 14, x^2 - 10x + 25, and x^2 + 2x - 35. Each example lists the steps to factor the expression which include finding factors of the constant term that sum to the coefficient of the middle term and then factoring the first term.
Solving by factoring remediation notes
Here are the key steps to factoring when a ≠ 1:
1. Check for GCF. If there is a GCF, factor it out.
2. If no GCF, find two numbers whose product is ac and sum is b. These will be the coefficients of the factors.
3. Split the middle term into two terms using the factors.
4. Factor using the box method, finding the GCF of each row and column.
Some examples:
2x2 + 11x + 5
ac = 10, factors are 2, 5
(2x + 1)(x + 5)
3x2 - 5x - 2
ac = -
Factoring Quadratic Trinomials
The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.
Factoring by grouping
How to factor by grouping
Assessments and examples are available
Can be used by a math teacher in her discussion about factoring
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factoring polynomials
This document outlines key concepts and examples for factoring polynomials. It covers factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL multiplication. Sections include the greatest common factor, factoring trinomials of the forms x^2 + bx + c and ax^2 + bx + c.
Factoring Polynomials to find its zeros
This document provides a lecture on factoring polynomials to find zeros. It begins with the lecture objectives of learning how to find the greatest common factor of a polynomial and factor simple quadratics to reveal their zeros. It then discusses finding the greatest common factor, factoring using the difference of squares formula, and factoring simple quadratics by finding two numbers that multiply and add to given values. Examples are provided for each technique. The document concludes by having students practice these skills by factoring and finding the zeros of polynomials.
Factoring polynomials
This document discusses factoring polynomials. It explains that factoring is reversing the process of multiplication to express a polynomial as a product of binomials. It provides examples of factoring out the greatest common factor (GCF) from a polynomial and factoring trinomials into two binomials. It outlines the step-by-step process of factoring trinomials, emphasizing the importance of checking signs. Finally, it provides examples for the reader to practice factoring polynomials.
Polynomials and factoring
The document discusses polynomials and factoring polynomials. It defines polynomials as expressions with terms added or subtracted, where terms are products of numbers and variables with exponents. It provides examples of monomials, binomials, trinomials, and polynomials based on the number of terms. It also discusses finding the greatest common factor of a polynomial to factor out a monomial.
Factoring quadratic trinomial
The document discusses factoring quadratic trinomials. It provides three examples of factoring expressions such as x^2 - 5x - 14, x^2 - 10x + 25, and x^2 + 2x - 35. Each example lists the steps to factor the expression which include finding factors of the constant term that sum to the coefficient of the middle term and then factoring the first term.
Solving by factoring remediation notes
Here are the key steps to factoring when a ≠ 1:
1. Check for GCF. If there is a GCF, factor it out.
2. If no GCF, find two numbers whose product is ac and sum is b. These will be the coefficients of the factors.
3. Split the middle term into two terms using the factors.
4. Factor using the box method, finding the GCF of each row and column.
Some examples:
2x2 + 11x + 5
ac = 10, factors are 2, 5
(2x + 1)(x + 5)
3x2 - 5x - 2
ac = -
Factoring Quadratic Trinomials
The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.
Factoring by grouping
How to factor by grouping
Assessments and examples are available
Can be used by a math teacher in her discussion about factoring
Multiplying Polynomials
1) The document discusses multiplying different types of polynomials, including: monomials, binomials, and polynomials with more than two terms.
2) It provides examples and steps for multiplying a monomial by a monomial, monomial by a polynomial, binomial by binomial, and a polynomial by a polynomial with three or more terms.
3) The key steps are distributing terms, applying rules like the distributive property, and combining like terms in the results.
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The document provides examples of factoring quadratic trinomials. It gives the steps to factor expressions such as 2x^2 - x - 6, 10x^2 + 3x - 1, and 3x^2 - x - 2. The steps involve multiplying the coefficient of x^2 by the constant term, finding the factors of the resulting number that sum to the coefficient of x, and then grouping the expression and factoring using the common factor.
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The document provides information about adding and subtracting algebraic expressions:
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- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
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1) The document discusses factoring trinomials of the form x^2 + bx + c by using the FOIL method in reverse.
2) It provides examples of factoring trinomials with positive and negative signs using FOIL, and introduces a sign flowchart to determine the signs of the binomial factors.
3) The key steps are to write the trinomial as a product of two binomials, look at the signs to determine the signs in the binomial factors, then guess and check the last two numbers to fully factor the trinomial.
Factorising Common Factors
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Grade 8 Mathematics Common Monomial Factoring
The document provides instructions for an individual activity where students must follow 16 directions within 2 minutes to complete tasks like writing their name, drawing shapes, and tapping their desk. It asks what implications the activity might have if directions are not followed properly. The second part of the document provides examples of factoring polynomials using greatest common factor and common monomial factoring methods. It includes practice problems for students to determine the greatest common factor, quotient, and factored form. The document emphasizes following directions and learning different factoring techniques.
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The document explains and provides examples of how to factor quadratic expressions using the "x-box" method. This method involves drawing an x-box with the coefficients of the expression and identifying the greatest common factor and two binomial factors that multiply to the original expression. Three examples are provided and worked through step-by-step to demonstrate how to set up and solve factoring problems using the x-box method.
Factoring by grouping ppt
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
3.2 factoring polynomials
This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.
Factoring
The document discusses various factoring techniques, including:
- Factoring by greatest common factor (GCF)
- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
- Factoring by grouping polynomials with four terms into two groups of two terms each.
Alg II Unit 4-4 Factoring Quadratic Expressions
This document provides instruction on factoring quadratic expressions. It begins with the essential understanding that many quadratic trinomials can be factored into products of two binomials. It then discusses factoring methods like finding common factors, using the distributive property, and factoring special forms like perfect square trinomials and differences of two squares. Examples are provided to demonstrate each factoring technique. The document concludes with assigning homework practice in factoring quadratic expressions.
Factorisation
This document provides an overview of factorizing algebraic expressions. It discusses several methods of factorizing including finding common factors, regrouping terms, using factorizing identities, and dividing polynomials. Examples are provided to illustrate each method. The document is divided into seven parts covering factors of natural numbers, common factors, regrouping terms, factorizing identities, factors in the form of (x+a)(x+b), and dividing algebraic expressions.
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This document discusses various methods for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF) of terms in a polynomial.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose product is c and sum is b.
3) Factoring polynomials using grouping when the GCF is not a single term.
4) Recognizing perfect square trinomials and factoring using (a + b)^2 = a^2 + 2ab + b^2.
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This document discusses manipulations of rational expressions. It begins by differentiating between rational numbers and rational expressions. The objectives are to simplify, multiply, divide, add and subtract rational expressions. Examples are provided to distinguish rational numbers from expressions. Steps are outlined for simplifying expressions, including factorizing and cancelling common factors. Students work through practice problems in groups and individually. The document concludes by assigning further study on multiplying and dividing rational expressions.
Factoring Polynomials
The document provides steps for factoring polynomials by finding the greatest common factor (GCF) and factoring binomials and trinomials. It explains how to factor expressions by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. Examples are provided to demonstrate how to factor polynomials into the product of binomials using factoring techniques.
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This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
GR 8 Math Powerpoint about Polynomial Techniques
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2. Three examples are shown applying these steps: finding the solution set of x^2 - 18x = -17 to be x = 17 and x = 1; solving x^2 + 10x - 11 = 0 to get x = 1 and x = -11; and solving x^2 = -24 + 10x to get x = 6 and x = 4.
Sketching the graph of a polynomial function
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Factoring GCF and Grouping
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
Factoring and Box Method
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
Polynomials
The document provides information about polynomials at Higher level, including:
- Definitions of polynomials and examples of polynomial expressions
- Evaluating polynomials using substitution and nested/synthetic methods
- The factor theorem and using it to factorize polynomials
- Finding missing coefficients in polynomials
- Finding the polynomial expression given its zeros
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1) The document discusses multiplying different types of polynomials, including: monomials, binomials, and polynomials with more than two terms.
2) It provides examples and steps for multiplying a monomial by a monomial, monomial by a polynomial, binomial by binomial, and a polynomial by a polynomial with three or more terms.
3) The key steps are distributing terms, applying rules like the distributive property, and combining like terms in the results.
Factoring if a is greater than 1 grade8
The document provides examples of factoring quadratic trinomials. It gives the steps to factor expressions such as 2x^2 - x - 6, 10x^2 + 3x - 1, and 3x^2 - x - 2. The steps involve multiplying the coefficient of x^2 by the constant term, finding the factors of the resulting number that sum to the coefficient of x, and then grouping the expression and factoring using the common factor.
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The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
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1) The document discusses factoring trinomials of the form x^2 + bx + c by using the FOIL method in reverse.
2) It provides examples of factoring trinomials with positive and negative signs using FOIL, and introduces a sign flowchart to determine the signs of the binomial factors.
3) The key steps are to write the trinomial as a product of two binomials, look at the signs to determine the signs in the binomial factors, then guess and check the last two numbers to fully factor the trinomial.
Factorising Common Factors
This document discusses finding the common factor of algebraic expressions. It explains that to find the common factor, one must break down all numbers within the expressions into their prime number factors. The common factors that are present in both expressions are then written outside of parentheses, while the remaining terms are written inside. Several examples are provided of factorizing expressions using this process of identifying common prime factors. The "highest common factor" refers to the largest common factor present outside of the parentheses.
Grade 8 Mathematics Common Monomial Factoring
The document provides instructions for an individual activity where students must follow 16 directions within 2 minutes to complete tasks like writing their name, drawing shapes, and tapping their desk. It asks what implications the activity might have if directions are not followed properly. The second part of the document provides examples of factoring polynomials using greatest common factor and common monomial factoring methods. It includes practice problems for students to determine the greatest common factor, quotient, and factored form. The document emphasizes following directions and learning different factoring techniques.
Diamond and box factoring student version
The document explains and provides examples of how to factor quadratic expressions using the "x-box" method. This method involves drawing an x-box with the coefficients of the expression and identifying the greatest common factor and two binomial factors that multiply to the original expression. Three examples are provided and worked through step-by-step to demonstrate how to set up and solve factoring problems using the x-box method.
Factoring by grouping ppt
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This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.
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The document discusses various factoring techniques, including:
- Factoring by greatest common factor (GCF)
- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
- Factoring by grouping polynomials with four terms into two groups of two terms each.
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This document provides instruction on factoring quadratic expressions. It begins with the essential understanding that many quadratic trinomials can be factored into products of two binomials. It then discusses factoring methods like finding common factors, using the distributive property, and factoring special forms like perfect square trinomials and differences of two squares. Examples are provided to demonstrate each factoring technique. The document concludes with assigning homework practice in factoring quadratic expressions.
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This document provides an overview of factorizing algebraic expressions. It discusses several methods of factorizing including finding common factors, regrouping terms, using factorizing identities, and dividing polynomials. Examples are provided to illustrate each method. The document is divided into seven parts covering factors of natural numbers, common factors, regrouping terms, factorizing identities, factors in the form of (x+a)(x+b), and dividing algebraic expressions.
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This document discusses various methods for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF) of terms in a polynomial.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose product is c and sum is b.
3) Factoring polynomials using grouping when the GCF is not a single term.
4) Recognizing perfect square trinomials and factoring using (a + b)^2 = a^2 + 2ab + b^2.
Simplifying Rational Expression
This document discusses manipulations of rational expressions. It begins by differentiating between rational numbers and rational expressions. The objectives are to simplify, multiply, divide, add and subtract rational expressions. Examples are provided to distinguish rational numbers from expressions. Steps are outlined for simplifying expressions, including factorizing and cancelling common factors. Students work through practice problems in groups and individually. The document concludes by assigning further study on multiplying and dividing rational expressions.
Factoring Polynomials
The document provides steps for factoring polynomials by finding the greatest common factor (GCF) and factoring binomials and trinomials. It explains how to factor expressions by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. Examples are provided to demonstrate how to factor polynomials into the product of binomials using factoring techniques.
Factorising quadratic expressions 1
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
GR 8 Math Powerpoint about Polynomial Techniques
-This is a powerpoint inspired by one of Canva displayed presentation.
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- It consists of 39 pages explaining each of the Polynomial Techniques.
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1. The document provides steps for solving quadratic equations using completing the square. It involves isolating terms with x, finding the square of half the coefficient of x, completing the square of the left side of the equation, applying the square root property, and solving for the roots.
2. Three examples are shown applying these steps: finding the solution set of x^2 - 18x = -17 to be x = 17 and x = 1; solving x^2 + 10x - 11 = 0 to get x = 1 and x = -11; and solving x^2 = -24 + 10x to get x = 6 and x = 4.
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This document discusses how to find the x-intercepts and y-intercepts of a polynomial function. To find the x-intercepts, one must factor the polynomial completely and set each factor equal to zero to solve for x. To find the y-intercept, set x equal to zero and simplify the polynomial. For the example polynomial function x^4 - 5x^2 + 4, the x-intercepts are (1,0), (-1,0), (2,0), (-2,0) and the y-intercept is (0,4).
Factoring GCF and Grouping
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
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1. The document discusses factoring polynomials. It covers factoring trinomials of the form x^2 + bx + c by finding two binomial factors with a sum of b and product of c.
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The document discusses various methods for factoring polynomials:
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Factoring polynomials
- 1. Factoring Polynomials NCVPS Summer 2010
- 2. Factoring is reversing the process of multiplying monomials, binomials or trinomials or any combination of these. We will focus on factoring out monomials (we call these GCFs) from a polynomial or factoring trinomials into two binomials. Ex: 3x(a – 3) or (x – 1)(x + 5) | | / monomial binomial binomials What is factoring?
- 3. Sometimes a common factor is present. Ex 1: 3x2 – 6x -9 has a common factor of 3: 3(x2 – 2x – 3) Ex 2: x3-5x2+4x has a common x in all 3 terms: x(x2 - 5x + 4) Ex 3: 2x2a + 6xa+12a has a common factor of 2a: 2a(x2 + 3x + 6) Step 1: Look for a GCF present
- 4. Factor first term into its products. Example 1: x2 – 2x – 3. The first term is x2. It is factored as x*x. Factor the last term into its products. The last term is -3. It factors as 1(-3) or (-1)(3) Step 2: Factor first & last terms
- 5. Example 1: x2 – 2x – 3. (x )(x ) Next, add the factors of -3: x2 – 2x – 3 (x -3)(x +1) Example 1 continued
- 6. It is very important to check your factoring to make sure you got the signs in the correct place. Multiply them out again! x2 - 2x – 3 = (x -3)(x +1) F: x(x) O: (-3)(1) = -3 I: -3x L: 1x Inner + Last terms = -2x Check the work!
- 7. Checking the signs is very important! x2 + 3x + 2 has all + signs. It factors with + signs: (x + 2)(x + 1) x2 – 6x + 5 has a +5 but the middle term is -. So 5 must factor as (-5)(-1): (x - 5)(x – 1) More examples on the next page Signs, signs, everywhere there’s signs!
- 8. X2 -2x – 8 has a -8 that factors as + and – Here we work with factors and sums. We want a sum of the factors to be -2 (the middle term) (-8)(1) = -8 sum the factors: -8+1 = -7 (8)(-1) = -8 sum the factors: 8 + (-1) = 7 (-4)(2) = -8 sum the factors: -4 + 2 = -2 This last one is the factorization we want! (x – 4)(x + 2) Keep on checking signs!
- 9. Find the factorization of x2 – 4x -12 -12 factors as (-3)(4) or (3)(-4) or (2)(-6) or (-2)(6) or (-1)(12) or (1)(-12). Which of these sums to -4? -3 + 4 = 1 3 + (-4) = -1 2 + (-6) = -4 We have a winner! x2 -4x – 12 is (x + 2)(x – 6) Another example
- 10. x2 – 2x – 15 2x2 – x – 3 3x2-12x +9 Go to the next slide for the solutions. You Try Problems
- 11. x2 – 2x – 15 = (x – 5)(x + 3) 2x2 – x – 3 = (2x - 3)(x +1) 3x2-12x +9 = 3(x2 – 4x + 3) = 3(x – 3)(x -1) You Try Problems