Factoring Special Polynomials


 Jacob Marshall
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1 6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These polynomials are special because they fit a recognizable pattern. Pattern recognition is an important element of mathematics. Many mathematical discoveries were made because somebody recognized a pattern. The first pattern, which we saw in Section 6.4, is called the difference of two squares. CAUTION Rules and Properties: The Difference of Two Squares What about the sum of two squares, such as x 2 25 In general, it is not possible to factor (using real numbers) a sum of two squares. So (x 2 25) (x 5)(x 5) a 2 b 2 (a b)(a b) (1) In words: The product of the sum and difference of two terms gives the difference of two squares. Equation (1) is easy to apply in factoring. It is just a matter of recognizing a binomial as the difference of two squares. To confirm this identity, use the FOIL method to multiply (a b)(a b) Example 1 Factoring the Difference of Two Squares NOTE We are looking for perfect squares the exponents must be multiples of 2 and the coefficients perfect squares 1, 4, 9, 16, and so on. (a) Factor x Note that our example has two terms a clue to try factoring as the difference of two squares. x 2 25 (x) 2 (5) 2 (x 5)(x 5) (b) Factor 9a a 2 16 (3a) 2 (4) 2 (3a 4)(3a 4) (c) Factor 25m 4 49n 2. 25m 4 49n 2 (5m 2 ) 2 (7n) 2 (5m 2 7n)(5m 2 7n) CHECK YOURSELF 1 Factor each of the following binomials. (a) y 2 36 (b) 25m 2 n 2 (c) 16a 4 9b 2 423
2 424 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS We mentioned earlier that factoring out a common factor should always be considered your first step. Then other steps become obvious. Consider Example 2. Example 2 Factoring the Difference of Two Squares Factor a 3 16ab 2. First note the common factor of a. Removing that factor, we have a 3 16ab 2 a(a 2 16b 2 ) We now see that the binomial factor is a difference of squares, and we can continue to factor as before. So a 3 16ab 2 a(a 4b)(a 4b) CHECK YOURSELF 2 Factor 2x 3 18xy 2. You may also have to apply the difference of two squares method more than once to completely factor a polynomial. Example 3 Factoring the Difference of Two Squares Factor m 4 81n 4. m 4 81n 4 (m 2 9n 2 )(m 2 9n 2 ) Do you see that we are not done in this case? Because m 2 9n 2 is still factorable, we can continue to factor as follows. NOTE The other binomial factor, m 2 9n 2, is a sum of two squares, which cannot be factored further. m 4 81n 4 (m 2 9n 2 )(m 3n)(m 3n) CHECK YOURSELF 3 Factor x 4 16y 4. NOTE Be sure you take the time to expand the product on the righthand side to confirm the identity. Two additional patterns for factoring certain binomials include the sum or difference of two cubes. Rules and Properties: The Sum or Difference of Two Cubes a 3 b 3 (a b)(a 2 ab b 2 ) (2) a 3 b 3 (a b)(a 2 ab b 2 ) (3)
3 FACTORING SPECIAL POLYNOMIALS SECTION Example 4 Factoring the Sum or Difference of Two Cubes NOTE We are now looking for perfect cubes the exponents must be multiples of 3 and the coefficients perfect cubes 1, 8, 27, 64, and so on. (a) Factor x The first term is the cube of x, and the second is the cube of 3, so we can apply equation (2). Letting a x and b 3, we have x 3 27 (x 3)(x 2 3x 9) (b) Factor 8w 3 27z 3. This is a difference of cubes, so use equation (3). 8w 3 27z 3 (2w 3z)[(2w) 2 (2w)(3z) (3z) 2 ] (2w 3z)(4w 2 6wz 9z 2 ) NOTE Again, looking for a common factor should be your first step. NOTE Remember to write the GCF as a part of the final factored form. (c) Factor 5a 3 b 40b 4. First note the common factor of 5b. The binomial is the difference of cubes, so use equation (3). 5a 3 b 40b 4 5b(a 3 8b 3 ) 5b(a 2b)(a 2 2ab 4b 2 ) CHECK YOURSELF 4 Factor completely. (a) 27x 3 8y 3 (b) 3a 4 24ab 3 In each example in this section, we factored a polynomial expression. If we are given a polynomial function to factor, there is no change in the ordered pairs represented by the function after it is factored. Example 5 Factoring a Polynomial Function Given the function f(x) 9x 2 15x, complete the following. (a) Find f(1). f(1) 9(1) 2 15(1) (b) Factor f(x). f(x) 9x 2 15x 3x(3x 5)
4 426 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS (c) Find f(1) from the factored form of f(x). f(1) 3(1)(3(1) 5) 3(8) 24 CHECK YOURSELF 5 Given the function f(x) 16x 5 10x 2, complete the following. (a) Find f(1). (c) Find f(1) from the factored form of f(x). (b) Factor f(x). CHECK YOURSELF ANSWERS 1. (a) (y 6)( y 6); (b) (5m n)(5m n); (c) (4a 2 3b)(4a 2 3b) 2. 2x(x 3y)( x 3y) 3. (x 2 4y 2 )(x 2y)( x 2y) 4. (a) (3x 2y)( 9x 2 6xy 4y 2 ); (b) 3a(a 2b)(a 2 2ab 4b 2 ) 5. (a) 26; (b) 2x 2 (8x 3 5); (c) 26
5 Name 6.6 Exercises Section Date For each of the following binomials, state whether the binomial is a difference of squares. 1. 3x 2 2y x 2 7y a 2 25b n 2 16m r p a 2 12b a 2 b 2 16c 2 d 2 9. a 2 b a 3 b 3 Factor the following binomials. 11. x m a b p x a m x 2 y m 2 n 2 9 ANSWERS c 2 25d a 2 49b p 2 64q x 2 36y x 4 16y a 2 25b
6 ANSWERS a 3 4ab p 2 q q a 4 16b x 4 y x y m b a 3 b p 3 q w 3 z c 3 27d r 3 64s x 3 y x 3 27y m 3 27n x 3 y m 6 27n x 3 32y a 3 81b x 3 2xy a 2 b 2b m 3 n 75mn p 4 7p 2 q 2 428
7 ANSWERS For each of the functions in exercises 51 to 56, (a) find f(1), (b) factor f(x), and (c) find f(1) from the factored form of f(x). 51. f(x) 12x 5 21x f(x) 6x 3 10x f(x) 8x 5 20x 54. f(x) 5x 5 35x f(x) x 5 3x f(x) 6x 6 16x 5 Factor each expression. 57. x 2 (x y) y 2 (x y) 58. a 2 (b c) 16b 2 (b c) 59. 2m 2 (m 2n) 18n 2 (m 2n) 60. 3a 3 (2a b) 27ab 2 (2a b) 61. Find a value for k so that kx 2 25 will have the factors 2x 5 and 2x Find a value for k so that 9m 2 kn 2 will have the factors 3m 7n and 3m 7n Find a value for k so that 2x 3 kxy 2 will have the factors 2x, x 3y, and x 3y Find a value for k so that 20a 3 b kab 3 will have the factors 5ab, 2a 3b, and 2a 3b Complete the following statement in complete sentences: To factor a number you Complete this statement: To factor an algebraic expression into prime factors means Verify the formula for factoring the sum of two cubes by finding the product (a b)(a 2 ab b 2 ). 68. Verify the formula for factoring the difference of two cubes by finding the product (a b)(a 2 ab b 2 ). 429
8 ANSWERS What are the characteristics of a monomial that is a perfect cube? 70. Suppose you factored the polynomial 4x 2 16 as follows: 4x 2 16 (2x 4)(2x 4) Would this be in completely factored form? If not, what would be the final form? Answers 1. No 3. Yes 5. No 7. No 9. Yes 11. (x 7)(x 7) 13. (a 9)(a 9) 15. (3p 1)(3p 1) 17. (5a 4)(5a 4) 19. (xy 5)(xy 5) 21. (2c 5d)(2c 5d) 23. (7p 8q)(7p 8q) 25. (x 2 4y)(x 2 4y) 27. a(a 2b)(a 2b) 29. (a 2 4b 2 )(a 2b)(a 2b) 31. (x 4)(x 2 4x 16) 33. (m 5)(m 2 5m 25) 35. (ab 3)(a 2 b 2 3ab 9) 37. (2w z)(4w 2 2wz z 2 ) 39. (r 4s)(r 2 4rs 16s 2 ) 41. (2x 3y)(4x 2 6xy 9y 2 ) 43. (2x y 2 )(4x 2 2xy 2 y 4 ) 45. 4(x 2y)(x 2 2xy 4y 2 ) 47. 2x(3x y)(3x y) 49. 3mn(2m 5n)(2m 5n) 51. (a) 33; (b) 3x 2 (4x 3 7); (c) (a) 12; (b) 4x( 2x 4 5); (c) (a) 4; (b) x 2 (x 3 3); (c) (x y) 2 (x y) 59. 2(m 2n)(m 3n)(m 3n)
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