1. Warm Up #2 Factor completely. 2. 2x2 – 5x – 3 1. x2 – x – 12
3. Use synthetic division to divide 2x3 – 3x2 – 18x – 8 by x – 4. 4 8 8 20 2 5 2

3. EXAMPLE 1
List possible rational zeros
List the possible rational zeros of f using the rational
zero theorem.
a. f (x) = x3 + 2x2 – 11x + 12
Factors of the constant term: + 1, + 2, + 3, + 4, + 6, + 12
Factors of the leading coefficient: + 1
Possible rational zeros: , , , , , + 1 2 3 4 6 12
Simplified list of possible zeros: + 1, + 2, + 3, + 4, + 6, + 12

4. EXAMPLE 1
List possible rational zeros
b f (x) = 4x4 – x3 – 3x2 + 9x – 10
Factors of the constant term: + 1, + 2, + 5, + 10
Factors of the leading coefficient: + 1, + 2, + 4
+ , , , , , , , , , , +
Possible rational zeros: 1 2 5 10 4
+ 10
Simplified list of possible zeros: + 1, + 2, + 5, + 10, +
, , + 5 2 1 4 +

5. GUIDED PRACTICE
for Example 1
List the possible rational zeros of f using the rational
zero theorem.
f (x) = x3 + 9x2 + 23x + 15
Factors of 15:
Factors of 1:
ANSWER
+ 1, + 3,

6. GUIDED PRACTICE
for Example 1
List the possible rational zeros of f using the rational
zero theorem.
2. f (x) =2x3 + 3x2 – 11x – 6
Factors of -6:
Factors of 2:

7. EXAMPLE 2
Find zeros when the leading coefficient is 1
Find all real zeros of f (x) = x3 – 8x2 + 11x + 20.
SOLUTION
Use the graphic display calculator
y1 = f(x)
y2 = 0

8. Zeros are 1 , 6 , -3 GUIDED PRACTICE for Example 2
Find all real zeros of the function.
f (x) = x3 – 4x2 – 15x + 18
Zeros are 1 , 6 , -3

9. GUIDED PRACTICE
for Example 2
Find all real zeros of the function.
f (x) = x3 – 8x2 + 5x+ 14

11. Factored form
Now solve the last quadratic for the imaginary solutions.
There are 5 solutions!

13. Class/Homework Assignment: Worksheet 5.6 ( #1 – 19 odd)