1. 6.5 The Remainder and Factor Theorems p. 352 How do you divide polynomials? What is the remainder theorem? What is the difference between synthetic substitution and synthetic division? What is the factor theorem? What is a zero of a polynomial function?

2. When you divide a Polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) with a remainder r(x) written: f(x) = q(x) + r(x) d(x) d(x)

3. The degree of the remainder must be less than the degree of the divisor!

4. Polynomial Long Division: You write the division problem in the same format you would use for numbers. If a term is missing in standard form …fill it in with a 0 coefficient. Example: 2x 4 + 3x 3 + 5x – 1 = x 2 – 2x + 2

5. 2x 4 = 2x 2 x 2 2x 2

6. +4x 2 -4x 3 2x 4 -( ) - 4x 2 7x 3 +5x 7x 3 = 7x x 2 +7x 7x 3 - 14x 2 +14x-( ) 10x 2 - 9x +10 10x 2 - 20x +20-( ) 11x - 21 remainder

7. The answer is written: 2x 2 + 7x + 10 + 11x – 21 x 2 – 2x + 2 Quotient + Remainder over divisor

8. Now you try one! y 4 + 2y 2 – y + 5 = y 2 – y + 1 Answer: y 2 + y + 2 + 3 y 2 – y + 1

9. Remainder Theorem: If a polynomial f(x) is divisible by (x – k), then the remainder is r = f(k). Now you will use synthetic division (like synthetic substitution) f(x)= 3x 3 – 2x 2 + 2x – 5 Divide by x - 2

10. f(x)= 3x 3 – 2x 2 + 2x – 5 Divide by x - 2 Long division results in ?...... 3x 2 + 4x + 10 + 15 x – 2 Synthetic Division: f(2) = 3-22-5 2 3 6 4 8 10 20 15 Which gives you: 3x 2 + 4x + 10 + 15 x-2

11. Synthetic Division Practice 1 Divide x 3 + 2x 2 – 6x -9 by (a) x-2 (b) x+3 (a) x-2 12-6-9 2 1 2 4 8 2 4 -5 Which is x 2 + 4x + 2 + -5 x-2

12. Synthetic Division Practice cont. (b) x+3 12-6-9 -3 1 -3 3 -3 9 0 x 2 – x - 3

13. Factor Theorem: A polynomial f(x) has factor x-k if f(k)=0 note that k is a ZERO of the function because f(k)=0

14. Factoring a polynomial Factor f(x) = 2x 3 + 11x 2 + 18x + 9 Given f(-3)=0 Since f(-3)=0 x-(-3) or x+3 is a factor So use synthetic division to find the others!!

15. Factoring a polynomial cont. 211189 -3 2 -6 5 -15 3 -9 0 (x + 3)(2x 2 + 5x + 3) So…. 2x 3 + 11x 2 + 18x + 9 factors to: Now keep factoring-- gives you: (x+3)(2x+3)(x+1)

16. Your turn! Factor f(x)= 3x 3 + 13x 2 + 2x -8 given f(-4)=0 (x + 1)(3x – 2)(x + 4)

17. Finding the zeros of a polynomial function f(x) = x 3 – 2x 2 – 9x +18. One zero of f(x) is x=2 Find the others! Use synthetic div. to reduce the degree of the polynomial function and factor completely. (x-2)(x 2 -9) = (x-2)(x+3)(x-3) Therefore, the zeros are x=2,3,-3!!!

18. Your turn! f(x) = x 3 + 6x 2 + 3x -10 X=-5 is one zero, find the others! The zeros are x=2,-1,-5 Because the factors are (x-2)(x+1)(x+5)

19. How do you divide polynomials? By long division What is the remainder theorem? If a polynomial f(x) is divisible by (x – k), then the remainder is r = f(k). What is the difference between synthetic substitution and synthetic division? It is the same thing What is the factor theorem? If there is no remainder, it is a factor. What is a zero of a polynomial function? One of the answers

20. Assignment Page 356, 15-53 odd