1. 2.5 Apply the Remainder and Factor Theorem
Pg. 85

2. Factor Theorem
(x – r) is a “factor” of the polynomial expression that defines the function P
if and only if……..
“r” is a solution of P(x) = 0, that is,
P(r) = 0.

3. Factor Theorem Use substitution to determine whether
(x + 2) is a factor of x3 – 2x2 – 5x + 6
If (x + 2) is a factor then x = - 2
(-2)3 – 2(-2)2 – 5(-2) + 6 =
(-8) – =
= 0 … yes (x + 2) is a factor

4. Dividing Polynomials
Dividing Polynomials to get the factors can be done in one of two ways
Long Division or Synthetic Division
Lets start with Long Division

5. Long Division
Now you must ask yourself, “what can I multiply “x” by to get “x3”
The answer is = x2
Binomial Divisor –
You must start with two terms in the dividend.

6. Factoring by Long Division Answer
Distribute Negative and Add
Remainder if there is any

7. Polynomial Long Division
All degrees of the polynomial must be represented, if not then placeholders must be used
Use zero (0) for each degree not represented
Divide by

8. Factoring by Synthetic Division (only with a Binomial)
(x-2) is a Factor…x=2 of
Drop 1st coefficient down
Use the Polynomials coefficients
Remainder if there is any
Coefficients of the quotient

9. Synthetic Division
All degrees of the polynomial must be represented, if not then placeholders must be used
Use zero (0) for each degree not represented
Since the divisor is a binomial we can use Synthetic Division
Divide by

10. Factoring a Polynomial
Factor given that
(x – 3) is a factor

11. Finding Zeros of a Function
Find the other zeros of given that f(-1) = 0.
Remember zeros are solutions or roots (x = )

12. Example (divide)

13. Homework
Pg. 87, 2 – 24 (even), 25
Pg. 88, 2 – 16 (even)