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Chapter 2

6. Chapter. Chapter 2. Factoring Polynomials. The Greatest Common Factor and Factoring by Grouping. Section. 6.1. Factoring Polynomials.

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Chapter 2

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  1. 6 Chapter Chapter 2 Factoring Polynomials

  2. The Greatest Common Factor and Factoring by Grouping Section 6.1

  3. Factoring Polynomials When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. The product is the factored form of the integer. When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial. The product is the factored form of the polynomial. • The process of writing a polynomial as a product is called factoring the polynomial.

  4. Finding the Greatest Common Factor of a List of Integers Objective 1

  5. Greatest Common Factor • Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved. Finding the GCF of a List of Integers 1. Write each number or polynomial as a product of prime factors. • 2. Identify common prime factors. • 3. Take the product of all common prime factors. If there are no common prime factors, GCF is 1.

  6. Example Find the GCF of each list of numbers. • a. 12 and 8 • 12 = 2 · 2· 3 • 8 = 2·2· 2 • So the GCF is 2·2 = 4. • b. 7 and 20 • 7 = 1 · 7 • 20 = 2 · 2 · 5 • There are no common prime factors so the GCF is 1.

  7. Example Find the GCF of each list of numbers. • a. 6, 8 and 46 • 6 = 2 · 3 • 8 = 2 · 2 · 2 • 46 = 2 · 23 • So the GCF is 2. • b. 144, 256 and 300 • 144 = 2 · 2 · 2 · 2 · 3 · 3 • 256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 • 300 = 2 · 2 · 3 · 5 · 5 • So the GCF is 2 · 2 = 4.

  8. Finding the Greatest Common Factor of a List of Terms Objective 2

  9. Example • a. x3 and x7 • x3 = x ·x·x • x7 = x ·x· x · x · x · x · x • So the GCF is x · x · x = x3 • b.6x5 and 4x3 • 6x5 = 2 · 3 · x · x · x • 4x3 = 2 · 2 · x · x · x • So the GCF is 2 · x · x · x = 2x3 Find the GCF of each list of terms.

  10. Example Find the GCF of the following list of terms. a3b2, a2b5 and a4b7 • a3b2 = a · a · a · b · b • a2b5 = a · a · b · b · b · b · b • a4b7 = a · a · a · a · b · b · b · b · b · b · b So the GCF is a · a · b · b = a2b2

  11. Helpful Hint Remember that the GCF of a list of terms contains the smallest exponent on each common variable. The GCF of x3y5, x6y4, and x4y6is x3y4. smallest exponent on y smallest exponent on x

  12. Factoring Out the Greatest Common Factor Objective 3

  13. Factoring Polynomials The first step in factoring a polynomial is to find the GCF of all its terms. Then we write the polynomial as a product by factoring out the GCF from all the terms. The remaining factors in each term will form a polynomial.

  14. Example Factor out the GCF in each of the following polynomials. a. 6x3 – 9x2 + 12x = 3x · 2 · x2 – 3x · 3 · x + 3x · 4 = 3x(2x2 – 3x + 4) b. 14x3y + 7x2y – 7xy = 7xy · 2x2 + 7xy · x – 7xy · 1 = 7xy(2x2 + x – 1)

  15. Example Factor out the GCF in each of the following polynomials. a. 6(x + 2) – y(x + 2) = 6 ·(x + 2)– y·(x + 2) • =(x + 2)(6 – y) b. xy(y + 1) – (y + 1) • = xy·(y + 1)– 1 ·(y + 1) • =(y + 1)(xy – 1)

  16. Factoring Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial. This will usually be followed by additional steps in the process.

  17. Factoring by Grouping Objective 4

  18. Factoring by Grouping Step 1: Group the terms in two groups so that each group has a common factor. Step 2: Factor out the GCF from each group. Step 3: If there is a common binomial factor, factor it out. Step 4: If not, rearrange the terms and try these steps again.

  19. Example Factor by grouping.

  20. Example Factor by grouping.

  21. Example Factor by grouping.

  22. Example Factor by grouping. 21x3y2 – 9x2y + 14xy – 6 = (21x3y2 – 9x2y) + (14xy – 6) = 3x2y(7xy – 3) + 2(7xy – 3) = (7xy – 3)(3x2 + 2)

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