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**Factoring Polynomials**

Chapter 4.1 Factoring Polynomials

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**The Greatest Common Factor**

11.1 The Greatest Common Factor

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**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.

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**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 or Terms Write each number or polynomial as a product of prime factors. Identify common prime factors. Take the product of all common prime factors. If there are no common prime factors, GCF is 1.

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**Greatest Common Factor**

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

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**Greatest Common Factor**

Example Find the GCF of each list of numbers. 6, 8 and 46 6 = 2 · 3 8 = 2 · 2 · 2 46 = 2 · 23 So the GCF is 2. 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.

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**Greatest Common Factor**

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

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**Greatest Common Factor**

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

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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 x smallest exponent on y

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**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.

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**Factoring out the GCF Example**

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

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**Factoring out the GCF Example**

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

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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. Example Factor y2 – 18x – 3xy2. y2 – 18x – 3xy2 = 3(30 + 5y2 – 6x – xy2) = 3(5 · · y2 – 6 · x – x · y2) = 3(5(6 + y2) – x (6 + y2)) = 3(6 + y2)(5 – x)

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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.

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**Factoring by Grouping Example Factor by grouping.**

21x3y2 – 9x2y + 14xy – 6 = (21x3y2 – 9x2y) + (14xy – 6) Group the terms. = 3x2y(7xy – 3) + 2(7xy – 3) Factor each group. = (7xy – 3)(3x2 + 2) Factor out the common factor, (7xy – 3).

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