1. Warm-ups Week /8/12
Find the zeros of f(x) = x3 + 2x2 – 13x + 10 algebraically (without a graphing calculator). What if I told you that one of the zeros is 2. Can you find the others without using your graphing calculator?

2. Lesson 2.3: Real Zeros of Polynomial Functions
There are two methods for dividing polynomials: long division and synthetic division. Synthetic division is quicker, but long division will work for for divisors of any form.

3. Long Division
Divide 2x4 + 4x3 – 5x2 + 2x – 3 by
x2 + 2x – 3.

4. Synthetic Division
Divide x4 – 10x2 + 2x by x – 3.

5. The Remainder and Factor Theorems
Remainder Theorem:
If a polynomial f(x) is divided by (x – k), then the remainder is r = f(k).
Factor Theorem:
A polynomial f(x) has a factor (x – k) if and only if f(k) = 0.

6. Example 1
Use the Remainder Theorem to find each remainder and state whether the binomial is a factor, and thus determines a root.
Is x - 1 a factor of 2x3 – 3x2 + x ?

7. Example 2
Divide 3x4 + 4x3 – 12 by x + 2.

8. Example 3
Divide –x6 + 4x4 + 3x2 + 2x by x + 2.

9. Example 4
-3 is a root for x3 + 5x2 – 2x – 24. Find the others.

10. The Rational Root Theorem
Sometimes it can be difficult to identify roots of a polynomial. We can use the Rational Root Theorem to give us direction in finding possible roots.
If you list the factors of the leading coefficient and the factors of the constant of a polynomial, the possible roots are the factors of the constant divided by the factors of the leading coefficient.

11. Examples
Find the possible rational roots. Then determine the rational roots.
x3 + x2 – 5x + 3
6x3 + 11x2 - 3x – 2
x3 + 8x2 + 16x + 5