Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Section 6.3 Factoring Trinomials of the form x 2 + bx + c

2 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Recall that factoring is the reverse process of multiplication. Using the FOIL method, we can show that (x – 3)(x – 8) = x 2 – 11x Therefore, by factoring we obtain x 2 – 11x + 24 = (x – 3)(x – 8). Trinomials of the Form x 2 + bx + c This trinomial results in the product of two binomials. The first term is x and the second term is a number (including its sign).

3 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 1.The answer will be of the form (x + m)(x + n). 2.m and n are numbers such that: a) When you multiply them, you get the last term, which is c. b) When you add them, you get the coefficient of the middle term, which is b. Factoring Trinomials of the Form x 2 + bx + c

4 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Example Factor. b 2 + 8b + 15 x 2 + 8x + 15 = (x + ?)(x + ?) Find two numbers that we can multiply together to get 15 and add together to get 8. The numbers are 3 and 5. x 2 + 8x + 15 = (x + 3)(x + 5) Check: (x + 3)(x + 5) = x 2 + 8x + 15

5 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Example Factor. Find two numbers that we can multiply together to get 20 and add together to get 12. Write down possible combinations. ProductSum 1, , , 59 This combination works.

6 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Example Factor. Find two numbers that we can multiply together to get  42 and add together to get 1. One factor will be positive and one will be negative. The numbers are 7 and  6.

7 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Example Factor. a. b. c.

8 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. The two numbers m and n will have the same signs if the last term of the polynomial is positive. 1. They will both be positive if the coefficient of the middle term is positive. 2. They will both be negative if the coefficient of the middle term is negative. Facts About Factoring Trinomials of the Form x 2 + bx + c x 2 + bx + c = (x m)(x n) x 2 + 5x + 6 = (x + 2)(x + 3) x 2  5x + 6 = (x  2)(x  3)

9 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. The two numbers m and n will have opposite signs if the last term is negative. 1. The larger of the absolute values of the two numbers will be given a plus sign if the coefficient of the middle term is positive. 2. The larger of the absolute values of the numbers will be given a negative sign if the coefficient of the middle term is negative. Facts About Factoring Trinomials of the Form x 2 + bx + c x 2 + 6x  7 = (x + 7)(x  1) x 2  6x  7 = (x  7)(x + 1) x 2 + bx + c = (x m)(x n)

10 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Example Factor. All of the terms have a common factor of 2. Factor out the common factor.

11 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Example Factor. All of the terms have a common factor of 3. Factor out the common factor.