1. 1 What we will learn today…  How to divide polynomials and relate the result to the remainder and factor theorems  How to use polynomial division

2. Objective: 6.5 The Remainder and Factor Theorems 2 Dividing Polynomials 1. When you divide a polynomial, f(x) by a divisor, d(x), you get a quotient polynomial, q(x) and a remainder polynomial, r(x). We can write:

3. Objective: 6.5 The Remainder and Factor Theorems 3 How Do We Do This Division?  Long Division! Divide 2x 4 + 3x 3 + 5x – 1 by x 2 – 2x + 2

4. Objective: 6.5 The Remainder and Factor Theorems 4 The Answer  We write the answer as:

5. Objective: 6.5 The Remainder and Factor Theorems 5 You Try  Divide: y 4 + 2y 2 – y + 5 by y 2 – y + 1

6. Objective: 6.5 The Remainder and Factor Theorems 6 Remainder Theorem  If a polynomial f(x) is divided by x – k, then the remainder is r = f(k).  For instance if the remainder after dividing a polynomial by x-2 is 15, f(2) would also be 15.

7. Objective: 6.5 The Remainder and Factor Theorems 7 Synthetic Division  Divide x 3 + 2x 2 – 6x – 9 by x – 2  You Try! Divide the polynomial by x + 3

8. Objective: 6.5 The Remainder and Factor Theorems 8 Factor Theorem  A polynomial f(x) has a factor x – k if and only if f(k) = 0 (no remainder).  A number is called a zero of a function when it causes the function to evaluate to (or equal) zero. These also happen to be the “solutions”.

9. Objective: 6.5 The Remainder and Factor Theorems 9 Using Synthetic Substitution  Use synthetic substitution to find the factors of: f(x) = 2x 3 + 11x 2 + 18x + 9 given that f(-3) = 0.

10. Objective: 6.5 The Remainder and Factor Theorems 10 Finding Zeros of a Polynomial Function  We can use synthetic division to find the zeros of a function. Example:  One zero of f(x) = x 3 – 2x 2 – 9x + 18 is x=2. Find the other zeros of the function.

11. Objective: 6.5 The Remainder and Factor Theorems 11 You Try  One zero of f(x) = x 3 + 6x 2 + 3x – 10 is x=-5. Find the other zeros of the function.

12. Objective: 6.5 The Remainder and Factor Theorems 12 Homework  Page 356, 17, 23, 27, 35, 39, 41, 49, 53, 55, 59