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Factoring Polynomials and Solving Equations by Factoring Copyright © Cengage Learning. All rights reserved. 5 Section 5.1 Factoring Out the Greatest Common Factor; Factoring by Grouping Copyright © Cengage Learning. All rights reserved. Objectives 1 Identify the greatest common factor of two or more monomials. 2 Factor a polynomial containing a greatest common factor. 3 Factor a polynomial containing a negative greatest common factor. 4 Factor a polynomial containing a binomial greatest common factor. 3 Objectives 5 Factor a four-term polynomial using grouping. 4 Factoring Out the Greatest Common Factor; Factoring by Grouping We will reverse the operation of multiplication and show how to find the factors of a known product. The process of finding the individual factors of a product is called factoring. We will limit our discussion of factoring polynomials to those that factor using only rational numbers. 5 1. Identify the greatest common factor of two or more monomials 6 Identify the greatest common factor of two or more monomials Recall that a natural number greater than 1 whose only factors are 1 and the number itself is called a prime number. The prime numbers less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47 A natural number is said to be in prime-factored form if it is written as the product of factors that are prime numbers. 7 Identify the greatest common factor of two or more monomials To find the prime-factored form of a natural number, we can use a factoring tree. For example, to find the prime-factored form of 60, we proceed as follows: Solution 1 1. Start with 60. 2. Factor 60 as 6 10. 3. Factor 6 and 10. Solution 2 1. Start with 60. 2. Factor 60 as 4 15. 3. Factor 4 and 15. 8 Identify the greatest common factor of two or more monomials We stop when only prime numbers appear. In either case, the prime factorization of 60 are 2 2 3 5. Thus, the prime-factored form of 60 is 22 3 5. This illustrates the fundamental theorem of arithmetic, which states that there is exactly one prime factorization for any natural number greater than 1. 9 Greatest Common Factor of 2 Numbers Greatest Common Factor (GCF) of 2 Numbers: The largest natural number that divides each of these numbers E.g., The GCF of 42 and 60 is 6, because 6 is the largest natural number that divides each of these numbers: 42 60 ---- = 7 and ---- = 10 6 6 10 Identify the greatest common factor of two or more monomials 42 = 2 3 7 60 = 2 2 3 5 Algebraic monomials also can have a greatest common factor. The right sides of the equations show the prime factorizations of 6a2b3, 4a3b2, and 18a2b. 6a2b3 = 2 3 a a b b b 11 Identify the greatest common factor of two or more monomials 4a3b2 = 2 2 a a a b b 18a2b = 2 3 3 a a b Since all three of these monomials have one factor of 2, two factors of a, and one factor of b, the GCF is 2aab or 2a2b 12 Example Find the GCF of 10x3y2, 60x2y, and 30xy2. Solution: Find the prime factorization of each of the three monomial. 10x3y2 = 2 5 x x x y y 60x2y = 2 2 3 5 x x y 30xy2 = 2 3 5 x y y 13 Example – Solution cont’d List each common factor the least number of times it appears in any one monomial: 2, 5, x, and y. Find the product of the factors in the list: 2 5 x y = 10xy 14 2. Factor a polynomial containing a greatest common factor 15 Factor a polynomial containing a greatest common factor Recall that the distributive property provides a way to multiply a polynomial by a monomial. For example, To reverse this process and factor the product 6x3 – 9x2y, we can find the GCF of each term (which is 3x2) and then use the distributive property. 16 Factor a polynomial containing a greatest common factor This process is called factoring out the greatest common factor. Finding the Greatest Common Factor (GCF) 1. Identify the number of terms. 2. Find the prime factorization of each term. 3. List each common factor the least number of times it appears in any one term. 4. Find the product of the factors found in the list to obtain the GCF. 17 Example Factor: 12y2 + 20y Solution: To find the GCF, we find the prime factorization of 12y2 and 20y. 12y2 = 2 2 3 y y 20y = 2 2 5 y GCF = 4y 18 Example – Solution cont’d We can use the distributive property to factor out the GCF of 4y. 12y2 + 20y = 4y 3y + 4y 5 = 4y(3y + 5) Check by verifying that 4y(3y + 5) = 12y2 + 20y. 19 3. Factor a polynomial containing a negative greatest common factor 20 Factor a polynomial containing a negative greatest common factor It is often useful to factor –1 out of a polynomial, especially if the leading coefficient is negative. 21 Example 6 Factor –1 out of –a3 + 2a2 – 4. Solution: –a3 + 2a2 – 4 = (–1)a3 + (–1)(–2a2) + (–1)4 = –1(a3 – 2a2 + 4) Write each term with a factor of –1. Factor out the GCF, –1. = –(a3 – 2a2 + 4) Check: –(a3 – 2a2 + 4) = –a3 + 2a2 – 4 22 4. Factor a polynomial containing a binomial greatest common factor 23 Factor a polynomial containing a binomial greatest common factor If the GCF of several terms is a polynomial, we can factor out the common polynomial factor. For example, since a + b is a common factor of (a + b)x and (a + b)y, we can factor out the (a + b). (a + b)x + (a + b)y = (a + b)(x + y) We can check by verifying that (a + b)(x + y) = (a + b)x + (a + b)y. 24 Example Factor a + 3 out of (a + 3) + (a + 3)2. Solution: Recall that a + 3 is equal to (a + 3)1 and that (a + 3)2 is equal to (a + 3)(a + 3). We can factor out a + 3 and simplify. (a + 3) + (a + 3)2 = (a + 3) 1 + (a + 3) (a + 3) = (a + 3)[1 + (a + 3)] Factor out a + 3, the GCF. = (a + 3)(a + 4) Combine like terms. 25 5. Factor a four-term polynomial using grouping 26 Factor a four-term polynomial using grouping Suppose we want to factor ax + ay + cx + cy Although no factor is common to all four terms, there is a common factor of a in ax + ay and a common factor of c in cx + cy. In this case, we group the first two terms and group the last two terms. We can factor out the a from the first two terms and the c from the last two terms to obtain ax + ay + cx + cy = a(x + y) + c(x + y) = (x + y)(a + c) Factor out (x + y). 27 Factor a four-term polynomial using grouping We can check the result by multiplication. (x + y)(a + c) = ax + cx + ay + cy = ax + ay + cx + cy Thus, ax + ay + cx + cy factors as (x + y)(a + c). This type of factoring is called factoring by grouping. 28 Example Factor: 2c + 2d – cd – d 2. Solution: 2c + 2d – cd – d 2 = 2(c + d) – d(c + d) = (c + d)(2 – d) Factor out 2 from (2c + 2d) and –d from (–cd – d2). Factor out (c + d). Check: (c + d)(2 – d) = 2c – cd + 2d – d 2 = 2c + 2d – cd – d 2 29