1. Polynomial Division Objective: To divide polynomials by long division and synthetic division

2. What you should learn How to use long division to divide polynomials by other polynomials How to use synthetic division to divide polynomials by binomials of the form (x – k) How to use the Remainder Theorem and the Factor Theorem

3. 1. x goes into x 3 ? x 2 times. 2. Multiply (x-1) by x 2. 4. Bring down 4x. 5. x goes into 2x 2 ?2x times. 6. Multiply (x-1) by 2x. 8. Bring down -6. 9. x goes into 6x? 3. Change sign, Add. 7. Change sign, Add 6 times. 11. Change sign, Add. 10. Multiply (x-1) by 6.

4. Long Division. Check

5. Divide.

6. Long Division. Check

7. Example Check =

8. Division is Multiplication

9. The Division Algorithm If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exists a unique polynomials q(x) and r(x) such that Where r(x) = 0 or the degree of r(x) is less than the degree of d(x).

10. Synthetic Division Divide x 4 – 10x 2 – 2x + 4 by x + 3 10-10-24 -3 1 +9 3 1 -3 1

11. Long Division. 1-2-8 3 1 3 1 3 -5

12. The Remainder Theorem If a polynomial f(x) is divided by x – k, the remainder is r = f(k).

13. The Factor Theorem A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Show that (x – 2) and (x + 3) are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x – 18 27-4-27-18 +2 2 4 11 22 18 36 9 18 0

14. Example 6 continued Show that (x – 2) and (x + 3) are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x – 18 27-4-27-18 +2 2 4 11 22 18 36 9 18 -3 2 -6 5 -15 3 -9 0

15. Uses of the Remainder in Synthetic Division The remainder r, obtained in synthetic division of f(x) by (x – k), provides the following information. 1.r = f(k) 2.If r = 0 then (x – k) is a factor of f(x). 3.If r = 0 then (k, 0) is an x intercept of the graph of f.