2.5 REMAINDER AND FACTOR THEOREMS

Using theorems to factor polynomials

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.

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 = ______

Examples/Practice

1) Divide using synthetic division.

2) Divide

Factoring Completely

Example……..

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.

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