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Warm-Up Exercises
Factor completely.
1. x2 – x – 12
ANSWER
(x – 4)(x + 3)
2. 2x2 – 5x – 3
ANSWER
(x – 3)(2x + 1)
Warm-Up Exercises
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
Warm-Up1Exercises
EXAMPLE
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
1
1
1
1
1
1
Simplified list of possible zeros: + 1, + 2, + 3, + 4, + 6, + 12
List possible rational zeros
Warm-Up1Exercises
EXAMPLE
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 , + 1 , + 2 , + 5 , + 10 , + 1 , + 2 , + 5
4
2 2
4
2
2
4
1
1
1
1
+ 10
4
Simplified list of possible zeros: + 1, + 2, + 5, + 10, + 1
2
+ 5 ,+ 1 ,+ 5
2
4
4
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
List the possible rational zeros of f using the rational
zero theorem.
1.
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
1
1
1
1
Simplified list of possible zeros: + 1, + 3, + 5 + 15
Warm-Up
Exercises
GUIDED
PRACTICE
2.
for Example 1
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 , + 3
2
2
1
1
1
1
Simplified list of possible zeros: + 1, + 2, + 3, + 6 + 1
2
+ 3
2
Find zeros when the leading coefficient is 1
Warm-Up2Exercises
EXAMPLE
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 = + 1 , + 2 , + 4 , + 5 , + 10 , + 20
1
1
1
1
1
1
Find zeros when the leading coefficient is 1
Warm-Up2Exercises
EXAMPLE
STEP 2
Test these zeros using synthetic division.
Test x =1:
1 1 –8
11
1
–7
–7
4
1
Test x = –1:
–1
1
–8
1
–1
–9
11
20
4
24
↑
1 is not a zero.
20
9 20
20
0
↑
–1 is a zero
Find zeros when the leading coefficient is 1
Warm-Up2Exercises
EXAMPLE
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)
ANSWER
The zeros of f are –1, 4, and 5.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 2
Find all real zeros of the function.
3.
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
1
1
1
Simplified list of possible zeros: –3, 1 , 6
Warm-Up
Exercises
GUIDED
PRACTICE
4.
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=+ 1 ,+ 2,+ 7
1
1
1
Warm-Up
Exercises
GUIDED
PRACTICE
STEP 2
for Example 2
Test these zeros using synthetic division.
Test x =1:
1 1 –8
1
–7
1
5
14
–7 –2
–2 12
↑
Test x = –1:
–1
1
1
–8
5
14
–1
–9
9
14
14
0
↑
1 is not a zero.
–1 is a zero.
Warm-Up
Exercises
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)
ANSWER
The zeros of f are –1, 2, and 7.
Find zeros when the leading coefficient is not 1
Warm-Up3Exercises
EXAMPLE
Find all real zeros of f (x) =10x4 – 11x3 – 42x2 + 7x + 12.
SOLUTION
STEP 1 List the possible rational zeros of f :
+ 1 , + 2 , + 3 , + 4 , + 6 , + 12 , + 1 , + 3 , + 1
1
1
1
1
2
1
1
2
5
, + 2 , + 3 , + 4 , + 6 , + 12 , + 1 , + 3
5
5
5
5
5
10
10
Find zeros when the leading coefficient is not 1
Warm-Up3Exercises
EXAMPLE
STEP 2
Choose reasonable
values from the list
above to check using
the graph of the
function. For f , the
values
x = – 23 , x = 1 , x = 53 , and x = 12
2
5
are reasonable based on the graph shown
at the right.
Find zeros when the leading coefficient is not 1
Warm-Up3Exercises
EXAMPLE
STEP 3
Check the values using synthetic
division until a zero is found.
– 3 10 –11 –42
7
12
2
9 – 69
–15 39
4
2
21
10 – 26 – 3 23 –
4
2
1
2
10
– 11
– 42
10
–5
– 16
8 17 –12
– 34 24
0
7
12
–
↑
1
2 is a zero.
Find zeros when the leading coefficient is not 1
Warm-Up3Exercises
EXAMPLE
STEP 4
Factor out a binomial using the result of
the synthetic division.
f (x) = x + 1
2
(10x3 – 16x2 – 34x + 24)
Write as a product of
factors.
= x + 1 (2)(5x3 – 8x2 – 17x + 12) Factor 2 out of the
2
second factor.
= (2x +1)(5x3 – 8x2 – 17x +12)
Multiply the first
factor by 2.
Find zeros when the leading coefficient is not 1
Warm-Up3Exercises
EXAMPLE
STEP 5
Repeat the steps above for
g (x) = 5x3 – 8x2 – 17x + 12. 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 , + 2 , + 3 , + 4 , + 6 , + 12
5
5
5
5
5
5
3
The graph of g shows that 5 may be a zero.
3
Synthetic division shows that 5 is a zero and
g (x) = x – 3 (5x2 – 5x – 20) = (5x – 3)(x2 – x – 4).
5
It follows that:
f (x) = (2x + 1)
g (x) = (2x + 1)(5x – 3)(x2 – x – 4)
Find zeros when the leading coefficient is not 1
Warm-Up3Exercises
EXAMPLE
STEP 6 Find the remaining zeros of f by solving
x2 – x – 4 = 0.
x = – (– 1) + √ (–
– 4(1)(– 4)
2(1)
1)2
x = 1 + √17
2
Substitute 1 for a, 21
for b, and 24 for c in
the quadratic
formula.
Simplify.
ANSWER
The real zeros of f are – 1
2
,
3 , 1 + √17, and 1 – √17.
5
2
2
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 3
Find all real zeros of the function.
5.
f (x) = 48x3+ 4x2 – 20x + 3
ANSWER
3 1 1
4 , 6 , 2
Warm-Up
Exercises
GUIDED
PRACTICE
6.
for Example 3
f (x) = 2x4 + 5x3 – 18x2 – 19x + 42
ANSWER
3
2, 2 , 1, +2 √2
Warm-Up4Exercises
EXAMPLE
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?
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
SOLUTION
STEP 1
Write an equation for the volume of the
ice sculpture.
1
4 = 3 x2(x + 1)
12 = x3 + x2
0 = x3 + x2 – 12
Write equation.
Multiply each side by 3 and simplify.
Subtract 12 from each side.
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
STEP 2
List the possible rational solutions:
+ 1 , + 2 , + 3 , + 4 ,+ 6 , +12
1
1
1
1
1
1
STEP 3
Test possible solutions. Only positive xvalues make sense.
1
1
1
1
0
– 12
1
2
2
2
2
– 10
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
2
1
1
0
– 12
1
2
3
6
6
12
0
↑
2 is a solution.
STEP 4 Check for other solutions. The other two
solutions, which satisfy
x2 + 3x + 6 = 0, are x = – 3 + i √ 15 and can be
2
discarded because they are imaginary
numbers.
Solve a multi-step problem
Warm-Up4Exercises
EXAMPLE
ANSWER
The only reasonable solution is x = 2. The base of the
mold is 2 feet by 2 feet.
The height of the mold is 2 + 1 = 3 feet.
Warm-Up
Exercises
GUIDED
PRACTICE
7.
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?
ANSWER
Base side: 3 ft, height: 2 ft
Warm-Up
Exercises
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
Warm-Up
Exercises
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 = x3 + 1 x2(6 – x). If
2
the volume of the shed is 80 cubic feet, find x.
ANSWER
4