H Algebra 2/Trig · Unit 3
A factor (x − c) means f(c) = 0. Use that equation to find the missing constant.
Factor theorem
If (x − c) is a factor of f(x), then f(c) = 0.
A factor such as (2x − 1) means c = 1/2.
The factor is x − c.
If (x − 2) is a factor, then f(2) = 0.
f(2) = (2)³ + 3(2)² + k(2) − 4
= 8 + 12 + 2k − 4
16 + 2k = 0
2k = −16
Divide both sides by 2.
k = −8
Answer: k = −8
If (x + 1) is a factor, then f(−1) = 0.
f(−1) = (−1)³ − 4(−1)² + k(−1) + 6
= −1 − 4 − k + 6
1 − k = 0
−k = −1
Multiply both sides by −1.
k = 1
Answer: k = 1
If (x − 3) is a factor, then f(3) = 0.
f(3) = (3)³ + k(3)² − 5(3) + 6
= 27 + 9k − 15 + 6
18 + 9k = 0
9k = −18
Divide both sides by 9.
k = −2
Answer: k = −2
If (x + 2) is a factor, then f(−2) = 0.
f(−2) = (−2)³ + 5(−2)² + k(−2) − 4
= −8 + 20 − 2k − 4
8 − 2k = 0
−2k = −8
Multiply both sides by −1.
2k = 8
Divide both sides by 2.
k = 4
Answer: k = 4
If (x − 1) is a factor, then f(1) = 0.
f(1) = (1)³ + 2(1)² + k(1) − 8
= 1 + 2 + k − 8
−5 + k = 0
k = 5
Answer: k = 5
If (x + 4) is a factor, then f(−4) = 0.
f(−4) = (−4)³ − (−4)² + k(−4) + 16
= −64 − 16 − 4k + 16
−64 − 4k = 0
−4k = 64
Multiply both sides by −1.
4k = −64
Divide both sides by 4.
k = −16
Answer: k = −16
The factor is not monic, so c is a fraction.
(2x − 1) = 0 gives x = 12.
If (2x − 1) is a factor, then f(12) = 0.
f(12) = 4(12)³ + k(12) − 1
= 12 + 12k − 1
−12 + 12k = 0
12k = 12
Multiply both sides by 2.
k = 1
Answer: k = 1
(2x + 1) = 0 gives x = −12.
If (2x + 1) is a factor, then f(−12) = 0.
f(−12) = 2(−12)³ + k(−12)² − (−12) + 2
= −14 + 14k + 12 + 2
94 + 14k = 0
14k = −94
Multiply both sides by 4.
k = −9
Answer: k = −9
(3x − 1) = 0 gives x = 13.
If (3x − 1) is a factor, then f(13) = 0.
f(13) = 9(13)³ + k(13) − 2
= 13 + 13k − 2
−53 + 13k = 0
13k = 53
Multiply both sides by 3.
k = 5
Answer: k = 5
(2x − 3) = 0 gives x = 32.
If (2x − 3) is a factor, then f(32) = 0.
f(32) = 2(32)³ + k(32)² + (32) − 6
= 274 + 94k + 32 − 6
94 + 94k = 0
94k = −94
Multiply both sides by 4.
9k = −9
Divide both sides by 9.
k = −1
Answer: k = −1
(4x + 1) = 0 gives x = −14.
If (4x + 1) is a factor, then f(−14) = 0.
f(−14) = 8(−14)³ + k(−14) + 1
= −18 − 14k + 1
78 − 14k = 0
−14k = −78
Multiply both sides by 8.
−2k = −7
Multiply both sides by −1.
2k = 7
Divide both sides by 2.
k = 72
Answer: k = 72
(2x + 3) = 0 gives x = −32.
If (2x + 3) is a factor, then f(−32) = 0.
f(−32) = 4(−32)³ + k(−32) − 3
= −272 − 32k − 3
−332 − 32k = 0
−32k = 332
Multiply both sides by 2.
−3k = 33
Multiply both sides by −1.
3k = −33
Divide both sides by 3.
k = −11
Answer: k = −11
k appears in two terms.
If (x + 1) is a factor, then f(−1) = 0.
f(−1) = (−1)⁴ − 3(−1)³ + k(−1)² + 5(−1) + k
= 1 + 3 + k − 5 + k
−1 + 2k = 0
2k = 1
Divide both sides by 2.
k = 12
Answer: k = 12
If (x − 1) is a factor, then f(1) = 0.
f(1) = (1)³ + k(1)² − 4(1) + k
= 1 + k − 4 + k
−3 + 2k = 0
2k = 3
Divide both sides by 2.
k = 32
Answer: k = 32
If (x − 2) is a factor, then f(2) = 0.
f(2) = (2)³ + k(2)² + k(2) − 6
= 8 + 4k + 2k − 6
2 + 6k = 0
6k = −2
Divide both sides by 2.
3k = −1
Divide both sides by 3.
k = −13
Answer: k = −13
If (x + 1) is a factor, then f(−1) = 0.
f(−1) = 2(−1)³ + k(−1)² − (−1) + 2k
= −2 + k + 1 + 2k
−1 + 3k = 0
3k = 1
Divide both sides by 3.
k = 13
Answer: k = 13
If (x + 2) is a factor, then f(−2) = 0.
f(−2) = (−2)⁴ + k(−2)³ − 2(−2)² + k(−2) + 6
= 16 − 8k − 8 − 2k + 6
14 − 10k = 0
−10k = −14
Multiply both sides by −1.
10k = 14
Divide both sides by 2.
5k = 7
Divide both sides by 5.
k = 75
Answer: k = 75
If (x − 1) is a factor, then f(1) = 0.
f(1) = (1)⁴ + 2(1)³ + k(1)² − 6(1) + k
= 1 + 2 + k − 6 + k
−3 + 2k = 0
2k = 3
Divide both sides by 2.
k = 32
Answer: k = 32
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