1. Section 3 – The Remainder and Factor Theorems
Chapter 4
Section 3 – The Remainder and Factor Theorems

2. Remainder Theorem
The Remainder Theorem: is that the remainder of any polynomial divided by the binomial factor (x-r) gives the value of the polynomial at x=r.
The Factor Theorem: is just a special case of this, which says if there is NO remainder, than (x-r) is a factor of the polynomial (since no remainder means that r is a zero).

3. Synthetic Division
Synthetic Division is where we divide a polynomial by a binomial.
EX 1: Divide x3 + 4x2 - 3x – 5 by x+3.
The way we set this up is similar to long division with numbers.
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Start w/ root on left coefficients on right
ONLY WORKS FOR linear binomial factors!

4. More EX 2: Use synthetic division to divide x3 – x2 + 2 by x+1.
EX 3: Use synthetic division to find out if x-1 is a factor of 2x3 – 3x2 + x
EX 4: Find the value of k so that the remainder of (x3 + 3x2 – kx – 24)/(x+3) is zero.

5. Finding Binomial Factors
EX 5: Determine the binomial factors of x3 + x2 – 4x – 4
(this question is a lead-in to Rational Roots Theorem and Descartes’ Rule of Signs! Remember them???)

6. Assignment
Chapter 4, Section 3
pgs #12-36E,45