1. Factor completely.
1. x2 – x – 12
ANSWER
(x – 4)(x + 3)
2. 2x2 – 5x – 3
ANSWER
(x – 3)(2x + 1)

2. Factor completely.
3. Use synthetic division to divide 2x3 – 3x2 – 18x – 8 by x – 4.
ANSWER
2x2 + 5x + 2 4.
The volume of a box is modeled by f (x) = x (x – 1)(x – 2), where x is the length in meters. What is the volume when the length is 3 meters.
ANSWER
6m3

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
SOLUTION
Factors of the constant term: + 1, + 3, + 5, + 15
Factors of the leading coefficient: + 1
Possible rational zeros: , , , + 1 3 5 15
Simplified list of possible zeros: + 1, + 3,

6. GUIDED PRACTICE
for Example 1
2. f (x) =2x3 + 3x2 – 11x – 6
SOLUTION
Factors of the constant term: + 1, + 2, + 3
Factors of the leading coefficient: + 1, + 2, + 4
Possible rational zeros:
+ , , , , , + 1 2 3 6 1
Simplified list of possible zeros: + 1, + 2, + 3, + 6 + 3 2 +

7. EXAMPLE 2
Find zeros when the leading coefficient is 1
Find all real zeros of f (x) = x3 – 8x2 +11x + 20.
SOLUTION
STEP 1
List the possible rational zeros. The leading coefficient is 1 and the constant term is 20. So, the possible rational zeros are:
x = , , , , , + 5 1 4 2 10 20

8. ↑ ↑ EXAMPLE 2 Find zeros when the leading coefficient is 1 STEP 2
Test these zeros using synthetic division. –
Test x =1:
1 – –
1 is not a zero. ↑
Test x = –1:
– – – –
–1 is a zero ↑

9. EXAMPLE 2
Find zeros when the leading coefficient is 1
Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 20).
STEP 3
Factor the trinomial in f (x) and use the factor theorem.
f (x) = (x + 1) (x2 – 9x + 20)
= (x + 1)(x – 4)(x – 5)
The zeros of f are –1, 4, and 5.
ANSWER

10. GUIDED PRACTICE
for Example 2
Find all real zeros of the function.
f (x) = x3 – 4x2 – 15x + 18
SOLUTION
Factors of the constant term: + 1, + 2, + 3, + 6, + 9
Factors of the leading coefficient: + 1
Possible rational zeros: , 1 3 6
Simplified list of possible zeros: –3, 1 , 6

11. GUIDED PRACTICE
for Example 2
f (x) + x3 – 8x2 + 5x+ 14
SOLUTION
STEP 1
List the possible rational zeros. The leading coefficient is 1 and the constant term is 14. So, the possible rational zeros are:
x = , , + 7 1 2

12. ↑ ↑ GUIDED PRACTICE for Example 2 STEP 2
Test these zeros using synthetic division. –
Test x =1:
1 –7 –2
–7 –
1 is not a zero. ↑
Test x = –1:
– – – –
–1 is a zero. ↑

13. GUIDED PRACTICE
for Example 2
Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 14).
STEP 3
Factor the trinomial in f (x) and use the factor theorem.
f (x) = (x + 1) (x2 – 9x + 14)
= (x + 1)(x + 4)(x – 7)
The zeros of f are –1, 2, and 7.
ANSWER

14. EXAMPLE 3
Find zeros when the leading coefficient is not 1
Find all real zeros of f (x) =10x4 – 11x3 – 42x2 + 7x + 12.
SOLUTION
STEP 1 6 3 2 1 12 +
, , , , , , , +
, + 4 5 10
+ , , , , , , , +
List the possible rational zeros of f :

15. EXAMPLE 3
Find zeros when the leading coefficient is not 1
STEP 2
x = – , x = , x = , and x = 3 2 12 5 1
are reasonable based on the graph shown at the right.
Choose reasonable values from the list above to check using the graph of the function. For f , the values

16. ↑ EXAMPLE 3 Find zeros when the leading coefficient is not 1 STEP 3
Check the values using synthetic division until a zero is found. 2 3
– –11 –
– – 9 69 4
– – – 23 21
– –
– –12
– – 2 1 1
is a zero. 2 – ↑

17. Find zeros when the leading coefficient is not 1
EXAMPLE 3
Find zeros when the leading coefficient is not 1
STEP 4
Factor out a binomial using the result of the synthetic division.
f (x) = x (10x3 – 16x2 – 34x + 24) 1 2
Write as a product of
factors.
= x (2)(5x3 – 8x2 – 17x + 12) 1 2
Factor 2 out of the
second factor.
= (2x +1)(5x3 – 8x2 – 17x +12)
Multiply the first
factor by 2.

18. EXAMPLE 3
Find zeros when the leading coefficient is not 1
Repeat the steps above for
g (x) = 5x3 – 8x2 – 17x Any zero of g will also be a zero of f. The possible rational zeros of g are:
x = + 1, + 2, + 3, + 4, + 6, + 12, , , , , , + 1 5 2 3 4 6 12
STEP 5
The graph of g shows that may be a zero. Synthetic division shows that is a zero and 3 5
g (x) = x – (5x2 – 5x – 20) 3 5
= (5x – 3)(x2 – x – 4).
f (x) = (2x + 1) g (x)
It follows that:
= (2x + 1)(5x – 3)(x2 – x – 4)

19. Find zeros when the leading coefficient is not 1
EXAMPLE 3
Find zeros when the leading coefficient is not 1
STEP 6
Find the remaining zeros of f by solving x2 – x – 4 = 0.
Substitute 1 for a, 21 for b, and 24 for c in the quadratic formula.
x = – (– 1) + √ (– 1)2 – 4(1)(– 4)
2(1)
x = 1 + √17 2
Simplify.
ANSWER 1 3
The real zeros of f are , , 1 + √17, and 1 – √17. – 2 5 2 2

20. GUIDED PRACTICE
for Example 3
Find all real zeros of the function.
f (x) = 48x3+ 4x2 – 20x + 3 1 2 3 4 6
, ,
ANSWER

21. GUIDED PRACTICE
for Example 3
f (x) = 2x4 + 5x3 – 18x2 – 19x + 42 3 2
2, , 1, +2 √2
ANSWER

22. EXAMPLE 4
Solve a multi-step problem
ICE SCULPTURES
Some ice sculptures are made
by filling a mold with water and
then freezing it. You are
making such an ice sculpture
for a school dance. It is to be
shaped like a pyramid with a
height that is 1 foot greater
than the length of each side of
its square base. The volume of
the ice sculpture is 4 cubic feet. What are the dimensions of the mold?

23. Solve a multi-step problem
EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Write an equation for the volume of the ice sculpture.
4 = x2(x + 1) 1 3
Write equation.
12 = x3 + x2
Multiply each side by 3 and simplify.
0 = x3 + x2 – 12
Subtract 12 from each side.

24. EXAMPLE 4
Solve a multi-step problem
STEP 2
+ , , , , , + 1 2 3 4 6 12
List the possible rational solutions:
STEP 3
Test possible solutions. Only positive x-values make sense.
– 12
– 10

25. ↑ EXAMPLE 4 Solve a multi-step problem 2 1 1 0 – 12 2 6 12 1 3 6 0
– 12 ↑
2 is a solution.
STEP 4
x2 + 3x + 6 = 0, are x = and can be
discarded because they are imaginary
numbers. 2
Check for other solutions. The other two solutions, which satisfy
– 3 + i √ 15

26. EXAMPLE 4
Solve a multi-step problem
The only reasonable solution is x = 2. The base of the mold is 2 feet by 2 feet.
The height of the mold is = 3 feet.
ANSWER

27. GUIDED PRACTICE
for Example 4
WHAT IF? In Example 4, suppose the base of the ice sculpture has sides that are 1 foot longer than the height. The volume of the ice sculpture is 6 cubic feet. What are the dimensions of the mold? 7.
Base side: 3 ft, height: 2 ft
ANSWER

28. Daily Homework Quiz
1. List the possible rational zeros of
f(x) = x3 + 8x2 – x + 4.
ANSWER
+ 1, + 2, + 4
Find all real zeros of the functions
2. f(x) = x3 – 3x2 – 6x + 8.
ANSWER
– 2, 1, 4

29. Daily Homework Quiz
3. f(x) = 2x3 – 3x2 – 17x + 30.
ANSWER
– 3, 2, 5 2
4. The volume V of a storage shed with a triangular
roof can be modeled by V = x x2(6 – x). If
the volume of the shed is 80 cubic feet, find x. 1 2
ANSWER 4