1. Polynomial Division and the Remainder Theorem Section 9.4

2. Polynomial long division Divide 2x³ + 3x² - x + 1 by x + 2 x + 2 is the divisor The quotient will be here. 2x³ + 3x² - x + 1 is the dividend

3. Polynomial long division First divide the first term of the dividend, 2x³, by x (the first term of the divisor). This gives 2x². This will be the first term of the quotient.

4. Polynomial long division Now multiply 2x² by x + 2 and subtract

5. Polynomial long division Bring down the next term, -x.-x.

6. Polynomial long division Now divide –x², the first term of –x² - x, by x, the first term of the divisor which gives –x.–x.

7. Polynomial long division Multiply –x by x + 2 and subtract

8. Polynomial long division Bring down the next term, 1

9. Polynomial long division Divide x, the first term of x + 1, by x,x, the first term of the divisor which gives 1

10. Polynomial long division Multiply x + 2 by 1 and subtract

11. Polynomial long division The remainder is –1. The quotient is 2x² - x + 1 Also:

12. Example 1 Divide p(x) = -2x 3 – 7x 2 + 10x – 25 by x + 5 Q(x) -2x 2 + 3x – 5

13. Example 2: Divide p(x) = -8x 5 + 2x 4 + x 2 - 8 by 2x 2 – 1 Q(x) -4x 3 + x 2 – 2x + 1 r(x) = -2x – 7 or

14. Remainder Theorem When a polynomial f(x) of degree ≥ 1 is divided by x – c, the remainder is the constant f(c).

15. Example 3: Let p(x) be defined by p(x) = 2x 5 – 18x 3 + x – 11. Find the remainder when p(x) is divided by x – 3. x – c = x – 3 so c = 3 The remainder is p(3). P(3) = 2(3) 5 – 18(3) 3 + 3 – 11 = -8

16. Homework Pages 581 - 582 1, 2, 7 - 12