1. 5.5 Apply the Remainder and Factor Theorems

7. Divisor
Quotient
Remainder

8. When you divide a polynomial f(x) by a divisor d(x) you get a quotient polynomial q(x) and a remainder polynomial r(x).
Remainder Theorem:
If a polynomial f(x) is divided by x – k
Then the remainder is r = f(k)

9. Recall to evaluate a polynomial you could use direct substitution.
For example evaluate:
when x = 3.
And…by synthetic substitution: 3
x-value
ADD
105 6 18 30 10 35 98 6
Mult. By 3 2
f(3)=98

10. 2 0 -8 5 -7 6 18 30 105 3 2 6 10 35 98 This is equivalent to dividing
By x - 3
synthetic substitution: 6 18 30
105 3 2 6
x-value 10 35 98 or

11. Use synthetic division to divide:
By x - 2

12. Factor Theorem:
A polynomial f(x) has a factor x – k if and only if f(k)=0.
Factor
given that f(-3) = 0.
Because f(-3) = 0, you know that x – (-3) or x+3 is a factor of f(x). Use synthetic division to find the other factors:
Solution: