1. Polynomial and Synthetic Division #3

2. Common Core objectives: * Students will be able to use long division to divide polynomials by other polynomials. Use synthetic division to divide polynomials by binomials of the form (x – k). Use the Remainder Theorem and the Factor Theorem. * Teacher will instruct students to learn about the process of division. Learn about a couple of theorems to help in factoring and solving higher level polynomials

3. Now let’s look at another method to divide… Why??? Sometimes it is easier…

4. Synthetic Division Synthetic Division is a ‘shortcut’ for polynomial division that only works when dividing by a linear factor (x + b). It involves the coefficients of the dividend, and the zero of the divisor.

5. Synthetic Division The pattern for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.) most vertical pattern: ADD terms Diagonal pattern: MULTIPLY terms

6. Example Divide: Step 1:  Write the coefficients of the dividend in a upside-down division symbol. 156

7. Example Step 2:  Take the zero (or root) from the divisor, and write it on the left, x – 1 = 0, so the zero is 1. 1 5 6 1

8. Example Step 3:  Carry down the first coefficient. 1561 1

9. Example Step 4:  Multiply the zero by this number. Write the product under the next coefficient. 1561 1 1

10. Example Step 5:  Add. 1561 1 1 6

11. Example Step 6 etc.:  Repeat as necessary 1561 1 1 6 6 12

12. step 7 The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original. 1561 1 1 6 6 12

13. Using Synthetic Division Use synthetic division to divide x 4 – 10x 2 – 2x + 4 by x + 3. Solution: You should set up the array as follows. Note that a zero is included for the missing +x 3 term in the dividend.

14. Example Divide: Step 1:  Write the coefficients of the dividend in a upside-down division symbol. 234 2x 2 + 3x + 4 x - 1

15. Example Step 2:  Take the zero (or root) from the divisor, and write it on the left, x – 1 = 0, so the zero is 1. 2 3 4 1 2x 2 + 3x + 4 x - 1

16. Example Step 3:  Carry down the first coefficient. 2341 2 2x 2 + 3x + 4 x - 1

17. Example Step 4:  Multiply the zero by this number. Write the product under the next coefficient. 2341 2 2 2x 2 + 3x + 4 x - 1

18. Example Step 5:  Add. 2341 2 2 5 2x 2 + 3x + 4 x - 1

19. Example Step 6 etc.:  Repeat as necessary 2341 2 2 5 5 9 2x 2 + 3x + 4 x - 1

20. step 7 The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original. 2341 2 2 5 5 9

21. ****Lab Application, practice