1. 9-5 Factoring x 2 + bx + c

2.  Factoring is the inverse of multiplying. We are rewriting a polynomial as the product of 2 factors. Definition

3.  Remember when we multiplied, the “c” was what the two factors (or last terms) MULTIPLIED to.  The “b” was the “OI” of the FOIL process. This is what was ADDED together. So, we are looking for two numbers that when we multiply we get “c”, but when we add, we get “b”. Those will become our factors! Factoring x 2 + bx + c

4. Example #1 Factor: x 2 + 7x + 12 We are looking for two numbers that when we multiply we get 12, but when we add, we get 7. What are all the ways of getting 12? 1 · 12 2·62·6 3·43·4 Factoring x 2 + bx + c Which pair adds to 7? Finally, write the factors (x+3)(x+4)

5. Example #2 Factor: y 2 + 6y – 27 We are looking for two numbers that multiply to -27, but add to 6. What are all the ways of getting -27? -1 · 27 -3 · 9 3 · -9 1 ·-27 Factoring x 2 + bx + c Which pair adds to 6? Finally, write the factors (y-3)(y+9)

6. Example #3 Factor: p 2 – 2p – 15 We are looking for two numbers that multiply to -15, but add to -2. What are all the ways of getting -15? -1 · 15 -3 · 5 3 · -5 1 ·-15 Factoring x 2 + bx + c Which pair adds to -2? Finally, write the factors (p+3)(p-5)

7. Example #4 Factor: p 2 – 2rp – 15r 2 We are looking for two numbers that multiply to -15, but add to -2. What are all the ways of getting -15? -1 · 15 -3 · 5 3 · -5 1 ·-15 Factoring x 2 + bx + c Which pair adds to -2? Finally, write the factors (p+3r)(p-5r)

8. Example #5 Factor: k 2 – 13k + 12 We are looking for two numbers that multiply to 12, but add to -13. What are all the ways of getting 12? 1 · 12 2·62·6 3·43·4 but none add to -13 Factoring x 2 + bx + c Which pair adds to -13 if both are negatives? Finally, write the factors (k-1)(k-12)

9. Today’s Assignment

10. Box Method for Factoring x 2 + bx + c  Enter 1 st term and last term in the diagonal top left to bottom right. 1 st term last term

11. Box Method for Factoring x 2 + bx + c  Look at c, the last term (this is what the factors must multiply to) 1 st term last term

12. Box Method for Factoring x 2 + bx + c  b is what the factors must add to 1 st term last term

13. Box Method for Factoring x 2 + bx + c  So we look for 2 numbers that multiply to get c and add to get b and enter them into the other diagonals (don’t forget to include the variable.) 1 st term last termfactor

14. Box Method for Factoring x 2 + bx + c  Finally, we find the GCF of each row and column…those become the factors of x 2 + bx + c. 1 st term last termfactor GCF

15. Example #5  Factor: x 2 + 8x + 7

16. Box Method for Factoring x 2 + 8x + 7  Enter 1 st term and last term in the diagonal top left to bottom right. x2x2 7

17. Box Method for Factoring x 2 + 8x + 7  Find c (this is what the factors must multiply to) x2x2 7 c = 7

18. Box Method for Factoring x 2 + 8x + 7  b is what the factors must add to x2x2 7 c = 7

19. Box Method for Factoring x 2 + 8x + 7  So we look for 2 numbers that multiply to get 7 and add to get 8 and enter them into the other diagonals (don’t forget to include the variable.) x2x2 7 c = 7 1x 7x

20. Box Method for Factoring x 2 + 8x + 7  Finally, we find the GCF of each row and column…those become the factors of x 2 + 8x + 7. x2x2 7 c = 7 1x 7x x 1 x7 (x+1)(x+7)