1. 7.4 THE REMAINDER & FACTOR THEOREMS Objectives: The student will be able to… 1)evaluate functions using synthetic substitution 2)determine whether a binomial is a factor of a polynomial by using synthetic substitution.

2. Recall… Find P(-2) for P(x) = -x 3 + 4x 2 - 8x - 6

3. THE REMAINDER THEOREM... Where all you care about is the remainder! We can use the Remainder Theorem to evaluate values of a function.

4. Why use the remainder Theorem? Less possibility for arithmetic error leading to more correct answers! Synthetic Substitution is… Using synthetic division to evaluate a function

5. Example 1: Using synthetic division and the Remainder Theorem, find P(a) for P(x) = -x 3 + 4x 2 - 8x - 6 when a = -2

6. Example 2: Using synthetic division and the Remainder Theorem find P(a) for P(x) = 2x 3 + 4x 2 – 10x - 9 when a = 3

7. How can you show that a binomial is a factor of a polynomial? Is (x – 3) a factor of x 3 + 4x 2 – 15x – 18?

8. 3 | 1 4 -15 -18 3 21 18 1 7 6 0 If you wanted to find the other factors, you would use the depressed polynomial. x 2 + 7x + 6 If the remainder is zero, the binomial is a factor of the polynomial.

9. The Factor Theorem The binomial x – a is a factor of the polynomial f(x) if and only if f(a) = 0.

10. Example 3: Show x+3 is a factor of x 3 + 6x 2 –x – 30. Then find the remaining factors.

11. Example 4: What are the factors of x 3 + 4x 2 – 15x – 18?

12. Example 5