2.5 Remainder and factor Theorems

1. 2.5 Remainder and factor Theorems Using theorems to factor polynomials

2. Remainder Theorem • If a polynomial f(x) is divided by x-k, then the remainder r = f(k) • This is saying, when you divide (using synthetic division) by some factor k, the remainder is the same as what you would get when you substitute the value of k. • Synthetic Division and Synthetic Substitution are the same thing – plugging a value in gets the same thing as if you were to divide by that factor.

3. Synthetic Division • The divisor must be in the form x-k. • Examples: • If you were asked to divide by x+1, you would plug in or synthetically divide by -1 • x+1=0 then x=-1 • Divide by x-6, means to plug in or divide by +6 • x-6=0, then x=6 • Divide by x+3, means to plug in or divide by _______ • x+3 = 0, then x = ______

4. Examples/Practice • Divide using synthetic division. • Divide

5. Factoring Completely • Example……..

6. Factor Theorem • A polynomial f(x) has a factor x-k, if and only if f(k)=0. • Examples: • If f(3)=0, then x=3 is a zero and x-3 is a factor • If f(-2)=0, then x=-2 is a zero and x+2 is a factor • If f(13) does not equal zero, then 13 has a remainder • If f(-5)=0, then ______ is a zero and _____ is a factor.

7. Examples of Factor Theorem • Find the other zeros given f(2)=0 for: • Do we plug in 2 or -2 for synthetic division? • f(2) = 0, so x=2 is a zero and x-2 is a factor so we plug in x=2 in the synthetic division. • Practice pg 87 #14 and/or 18