Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 1 Chapter 6 Polynomial Functions

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide Factoring Trinomials of the Form x 2 + bx + c; Factoring Out the GCF

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 3 Comparing Multiplying with Factoring Multiplying and factoring are reverse processes. For example,

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 4 Factoring a Trinomial of the Form x 2 + bx + c To see how to factor x 2 + 5x + 6, let’s take another look at how we find the product (x + 2)(x + 3):

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 5 Example: Factoring a Trinomial of the Form x 2 + bx + c Factor x x + 24.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 6 Solution We need two integers whose product is 24 and whose sum is 11. Since 3(8) = 24 and = 11, we conclude that the last terms of the factors are 3 and 8.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 7 Solution x x + 24 = (x + 3)(x + 8) Check by finding the product of the result: (x + 3)(x + 8) = x 2 + 8x + 3x + 24 = x x + 24 By the commutative law, (x + 3)(x + 8) = (x + 8)(x + 3), so we can write the factors in either order.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 8 Factoring x 2 + bx + c To factor x 2 + bx + c, look for two integers p and q whose product is c and whose sum is b. That is pq = c and p + q = b. If such integers exist, the factored polynomial is (x + p)(x + q)

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 9 Factoring x 2 + bx + c with c Positive To factor a trinomial of the form x 2 + bx + c with a positive constant term c, If b is positive, look for two positive integers whose product is c and whose sum is b. For example,

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 10 Factoring x 2 + bx + c with c Positive If b is negative, look for two negative integers whose product is c and whose sum is b. For example,

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 11 Example: Factring a Trinomial of the Form x 2 + bx + c Factor w 2 – 3w – 18.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 12 Solution We need two integers whose product is –18 and whose sum is –3. Since the product is negative, the two integers must have difference signs. Here are the possibilities:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 13 Solution Since 3(–6) = –18 and 3 + (–6) = –3, we conclude that the last terms of the factors are 3 and –6. (w + 3)(w – 6) Check by finding the product: (w + 3)(w – 6) = w 2 – 6w + 3w – 18 = w 2 – 3w – 18

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 14 Factoring x 2 + bx + c with c Negative To factor a trinomial of the form x 2 + bx + c with a negative constant term c, look for two integers with different signs whose product is c and whose sum is b. For example,

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 15 Example: Factoring a Trinomial with Two Variables Factor a 2 + 6ab + 8b 2.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 16 Solution Write the trinomial in the form a 2 + (6b)a + 8b 2. We need two monomials whose product is 8b 2 and whose sum is 6b. So, the last two terms are 2b and 4b. a 2 + 6ab + 8b 2 = (a + 2b)(a + 4b) Check by finding the product. (a + 2b)(a + 4b) = a 2 + 6ab + 8b 2

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 17 Prime Polynomials Just as a prime number has no positive factors other than itself and 1, a polynomial that cannot be factored is called prime.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 18 Example: Identifying a Prime Polynomial Factor –14 + 6x + x 2.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 19 Solution Write the polynomial in descending order: x 2 + 6x – 14

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 20 Solution We need two integers whose product is –14 and whose sum is 6. Since the product is negative, the integers must have different signs. Here are the possibilities: Because none of the sums equal 6, we conclude that the trinomial is prime.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 21 Factoring Out the GCF Definition The greatest common factor (GCF) of two or more terms is the monomial with the largest coefficient and the highest degree that is a factor of all the terms.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 22 Example: Factoring Out the GCF Factor 18x 4 – 30x 2.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 23 Solution Begin by factoring 18x 4 and 30x 2 : 18x 4 = 2 ∙ 3 ∙ 3 ∙ x ∙ x ∙ x ∙ x 30x 2 = 2 ∙ 3 ∙ 5 ∙ x ∙ x There are four common factors, shown in blue. So, the GCF is 6x 2 :

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 24 Solution Use a graphing calculator table to verify our work.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 25 Example: Completely Factoring a Polynomial Factor 3x x x.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 26 Solution The GCF is 3x: 3x x x = 3x(x 2 + 7x + 12)

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 27 Solution Temporarily put aside the GCF, 3x. To factor x 2 + 7x + 12, we need two integers whose product is 12 and whose sum is 7: Because 3(4) = 12 and = 7, we conclude that the last terms of the factors are 3 and 4.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 28 Solution So,

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 29 Factor Out the GCF First In general, when the leading coefficient of a polynomial is positive and the GCF is not 1, first factor out the GCF. But, don’t forget to include the GCF as part of your answer!

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 30 Factoring Completely Warning When factoring a polynomial, always completely factor it.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 31 Factoring when the Leading Coefficient is Negative When the leading coefficient of a polynomial is negative, first factor out the opposite of the GCF.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 32 Example: Factoring Out the Opposite of the GCF Factor –2r r 3 – 40r 2.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 33 Solution For this polynomial, the GCF is 2r 2. The leading coefficient of –2r r 3 – 40r 2 is –2, which is negative. So, first factor out the opposite of the GCF:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 34 Solution Use a graphing calculator table to verify our work.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 35 Summary 1. If the leading coefficient of a polynomial is positive and the GCF is not 1, first factor out the GCF. If the leading coefficient is negative, first factor out the opposite of the GCF. 2. Always completely factor a polynomial.