Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 5 Polynomials and Factoring

5-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Factoring Trinomials of the Type x 2 + bx + c When the Constant Term Is Positive When the Constant Term Is Negative Prime Polynomials Factoring Completely 5.2

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. To Factor x 2 + bx + c when c is Positive When the constant term of a trinomial is positive, look for two numbers with the same sign. The sign is that of the middle term: x 2 – 7x + 10 (x – 2)(x – 5); x 2 + 7x + 10 (x + 2)(x + 5);

5-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Pairs of Factors of 12 Sums of Factors 1, , 68 3, 47  1,  12  13  2,  6 88  3,  4 77 Factor: x 2 + 7x + 12 Solution Think of FOIL in reverse. (x + )(x + ) We need a constant term that has a product of 12 and a sum of 7. We list some pairs of numbers that multiply to 12.

5-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Factor: x 2 + 7x + 12 Since 3  4 = 12 and = 7, the factorization of x 2 + 7x + 12 is (x + 3)(x + 4). To check we simply multiply the two binomials. Check: (x + 3)(x + 4) = x 2 + 4x + 3x + 12 = x 2 + 7x + 12

5-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Factor: y 2  8y + 15 Solution Since the constant term is positive and the coefficient of the middle term is negative, we look for the factorization of 15 in which both factors are negative. Their sum must be  8. Pairs of Factors of 15 Sums of Factors  1,  15  16  3,  5 88 Sum of  8 y 2  8y + 15 = (y  3)(y  5)

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. To Factor x 2 + bx + c When c is Negative When the constant term of a trinomial is negative, look for two numbers whose product is negative. One must be positive and the other negative: x 2 – 4x – 21 = (x + 3)(x – 7); x 2 + 4x – 21 = (x – 3)(x + 7). Select the two numbers so that the number with the larger absolute value has the same sign as b, the coefficient of the middle term.

5-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Factor: x 2  5x  24 Solution The constant term must be expressed as the product of a negative number and a positive number. Since the sum of the two numbers must be negative, the negative number must have the greater absolute value. x 2  5x  24 = (x + 3)(x  8) Pairs of Factors of  24 Sums of Factors 1,  24  23 2,  12  10 3,  8 55 4,  6 22 6,  4 2 8,  3 5

5-9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Factor: t 2  t Solution Rewrite the trinomial t 2 + 4t  32. We need one positive and one negative factor. The sum must be 4, so the positive factor must have the larger absolute value. t 2 + 4t  32 = (t + 8)(t  4) Pairs of Factors of  32 Sums of Factors  1,  2,  4, 8 4

5-10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Factor: a 2 + ab  30b 2 Solution We need the factors of  30b 2 that when added equal b. Those factors are  5b and 6b. a 2 + ab  30b 2 = (a  5b)(a + 6b)

5-11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Prime Polynomials A polynomial that cannot be factored is considered prime. Example: x 2  x + 7 Often factoring requires two or more steps. Remember, when told to factor, we should factor completely. This means the final factorization should contain only prime polynomials.

5-12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Factor: 2x 3  24x x Solution Always look first for a common factor. We can factor out 2x: 2x(x 2  12x + 36) Since the constant term is positive and the coefficient of the middle term is negative, we look for the factorization of 36 in which both factors are negative. Their sum must be  12.

5-13 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example The factorization of (x 2  12x + 36) is (x  6)(x  6) or (x  6) 2 The factorization of 2x 3  24x x is 2x(x  6) 2 or 2x(x  6)(x  6) Pairs of Factors of 36 Sums of Factors  1,  36  37  2,  18  20  3,  12  15  4,  9  13  6,  6  12

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. To Factor x 2 + bx + c 1. Find a pair of factors that have c as their product and b as their sum. a)If c is positive, its factors will have the same sign as b. b)If c is negative, one factor will be positive and the other will be negative. Select the factors such that the factor with the larger absolute value has the same sign as b. 2.Check by multiplying. Once common factors have been factored out, the following summary can be used.