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.

Recall…Find P(-2) for P(x) = -x3 + 4x2 - 8x - 6

THE REMAINDER THEOREM . . .Where all you care about is the remainder!

• We can use the Remainder Theorem to evaluate values of a function.

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

Example 1: Using synthetic division and the Remainder Theorem, find P(a) for P(x) = -x3 + 4x2 - 8x - 6 when a = -2

Example 2: Using synthetic division and the Remainder Theorem find P(a) for P(x) = 2x3 + 4x2 – 10x - 9 when a = 3

How can you show that a binomial is a factor of a polynomial?

Is (x – 3) a factor of x3 + 4x2 – 15x – 18?

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.

x2 + 7x + 6

If the remainder is zero, the binomial is a factor of the polynomial.

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

Example 3: Show x+3 is a factor of

x3 + 6x2 –x – 30. Then find the remaining factors.

Example 4: What are the factors of

x3 + 4x2 – 15x – 18?

Example 5