1. © 2010 Pearson Prentice Hall. All rights reserved Factoring Trinomials “Reverse FOIL”

2. Tobey & Slater, Intermediate Algebra, 6e 2 Trinomials of the Form x 2 + bx + c Using the FOIL method, we can show that (x – 3)(x – 8) x 2 – 8x – 3x + 24 x 2 – 11x + 24. Therefore, by factoring (“reverse FOIL”) we obtain x 2 – 11x + 24 = (x – 3)(x – 8). Recall that factoring is the reverse process of multiplication. Recognize this form! This trinomial results in the product of two binomials.

3. Tobey & Slater, Intermediate Algebra, 6e 3 Trinomials of the Form x 2 + bx + c Factoring Trinomials of the Form x 2 + bx + c the answer will be of the form (x + m)(x + n) Factoring Trinomials of the Form x 2 + bx + c the answer will be of the form (x + m)(x + n) nx + mx = bx nx mx x 2 + bx + c (x + m)(x + n) FIRST terms come from the FIRST term. LAST terms Come from The LAST term. x. x = x 2 Inside and outside terms combine to be the middle term. m. n = c( )( )(x )(x ( m)( n

4. Tobey & Slater, Intermediate Algebra, 6e 4 Trinomials of the Form x 2 + bx + c Example: Factor. x 2 + 8x + 15 (x + 3)(x + 5) Answer x 2 + 8x + 15 (x + 15)(x + 1) 1x 15x (x + 3)(x + 5) 5x 3x x. x 15 x 1 or 5 x 3 or

5. Tobey & Slater, Intermediate Algebra, 6e 5 Facts About Signs Suppose x 2 + bx + c = (x + m)(x + n). We know certain facts about m and n. Facts About Signs Suppose x 2 + bx + c = (x + m)(x + n). We know certain facts about m and n. 1. m and n have the same signs if c is positive. (Note: We did not say that they will have the same sign as c.) a)They are positive if b is positive. b)They are negative if b is negative. 2. m and n have opposite signs if c is negative. The larger number is positive if b is positive and negative if b is negative.

6. Tobey & Slater, Intermediate Algebra, 6e 6 Trinomials of the Form x 2 + bx + c Example: Factor. x 2 + x – 42 One factor will be positive and one will be negative. x 2 + x – 42 ( )() This form trinomial will factor into two binomials.

7. Tobey & Slater, Intermediate Algebra, 6e 7 Try this. Factor. x 2 – 9x + 8

8. Tobey & Slater, Intermediate Algebra, 6e 8 Trinomials of the Form ax 2 + bx + c Example: Factor. 3x 2 + 4x + 1 (3x + 1)(x + 1) 3x 2 + 4x + 1 3x.1x3x.1x 1.11.1 3x3x 1x1x 1x + 3x = 4x Answer

9. Tobey & Slater, Intermediate Algebra, 6e 9 Trial and Error method Example: Factor. 9x 2  13x + 4 1. 4 2. 2 (x – 1)(9x – 4) 1x. 9x 3x. 3x The signs in both factors will be negative. (x – 2)(9x – 2)(3x – 1)(3x – 4) (3x – 2)(3x – 2) – 13x – 20x – 15x – 12x Correct 9x 2  13x + 4

10. Tobey & Slater, Intermediate Algebra, 6e 10 Try this: Factor. a)8x 2 + 10x  3b) 4x 2 – 13x – 12