H Algebra 2/Trig · Unit 3
The remainder is f(c). Two remainders give two equations.
Remainder theorem
When f(x) is divided by (x − c), the remainder is f(c).
Two remainders
Each remainder gives an equation. Solve the system for the two missing constants.
One remainder, one constant.
The remainder theorem: the remainder when f(x) is divided by (x − 1) is f(1).
f(1) = 4
f(1) = (1)³ + 2(1)² + k(1) − 5
= 1 + 2 + k − 5
−2 + k = 4
k = 6
Answer: k = 6
The remainder theorem: the remainder when f(x) is divided by (x − 2) is f(2).
f(2) = 5
f(2) = (2)³ − 3(2) + k
= 8 − 6 + k
2 + k = 5
k = 3
Answer: k = 3
The remainder theorem: the remainder when f(x) is divided by (x + 1) is f(−1).
f(−1) = −3
f(−1) = 2(−1)³ + k(−1)² − 4
= −2 + k − 4
−6 + k = −3
k = 3
Answer: k = 3
The remainder theorem: the remainder when f(x) is divided by (x − 1) is f(1).
f(1) = 6
f(1) = (1)⁴ + k(1) − 2
= 1 + k − 2
−1 + k = 6
k = 7
Answer: k = 7
The remainder theorem: the remainder when f(x) is divided by (x + 2) is f(−2).
f(−2) = −7
f(−2) = (−2)³ + 4(−2)² + k(−2) + 1
= −8 + 16 − 2k + 1
9 − 2k = −7
−2k = −16
Multiply both sides by −1.
2k = 16
Divide both sides by 2.
k = 8
Answer: k = 8
The remainder theorem: the remainder when f(x) is divided by (x − 3) is f(3).
f(3) = 10
f(3) = (3)³ − k(3) + 4
= 27 − 3k + 4
31 − 3k = 10
−3k = −21
Multiply both sides by −1.
3k = 21
Divide both sides by 3.
k = 7
Answer: k = 7
The divisor gives a fraction value of x.
The remainder theorem: the remainder when f(x) is divided by (2x − 1) is f(12).
f(12) = 2
f(12) = 4(12)³ + k(12) − 3
= 12 + 12k − 3
−52 + 12k = 2
12k = 92
Multiply both sides by 2.
k = 9
Answer: k = 9
The remainder theorem: the remainder when f(x) is divided by (2x + 1) is f(−12).
f(−12) = 3
f(−12) = 8(−12)³ + k(−12) − 2
= −1 − 12k − 2
−3 − 12k = 3
−12k = 6
Multiply both sides by 2.
−k = 12
Multiply both sides by −1.
k = −12
Answer: k = −12
The remainder theorem: the remainder when f(x) is divided by (2x − 1) is f(12).
f(12) = 4
f(12) = (12)³ + k(12)² + 2
= 18 + 14k + 2
178 + 14k = 4
14k = 158
Multiply both sides by 8.
2k = 15
Divide both sides by 2.
k = 152
Answer: k = 152
The remainder theorem: the remainder when f(x) is divided by (2x + 1) is f(−12).
f(−12) = −1
f(−12) = 2(−12)³ − (−12) + k
= −14 + 12 + k
14 + k = −1
k = −54
Multiply both sides by 4.
4k = −5
Divide both sides by 4.
k = −54
Answer: k = −54
The remainder theorem: the remainder when f(x) is divided by (3x − 1) is f(13).
f(13) = 4
f(13) = 9(13)³ + k(13) + 1
= 13 + 13k + 1
43 + 13k = 4
13k = 83
Multiply both sides by 3.
k = 8
Answer: k = 8
The remainder theorem: the remainder when f(x) is divided by (2x − 3) is f(32).
f(32) = 8
f(32) = 4(32)³ + k(32) + 5
= 272 + 32k + 5
372 + 32k = 8
32k = −212
Multiply both sides by 2.
3k = −21
Divide both sides by 3.
k = −7
Answer: k = −7
Two missing constants.
Call these Equation 1 and Equation 2.
Divided by (x + 1), the remainder is f(−1).
f(−1) = 2
f(−1) = (−1)⁴ + p(−1)³ + q(−1) + 4
= 1 − p − q + 4
5 − p − q = 2
−p − q = −3
Multiply both sides by −1.
p + q = 3
Equation 1: p + q = 3
Divided by (x − 2), the remainder is f(2).
f(2) = 14
f(2) = (2)⁴ + p(2)³ + q(2) + 4
= 16 + 8p + 2q + 4
20 + 8p + 2q = 14
8p + 2q = −6
Divide both sides by 2.
4p + q = −3
Equation 2: 4p + q = −3
Subtract the first equation from the second.
3p = −6
Divide both sides by 3.
p = −2
Substitute p = −2.
−2 + q = 3
q = 5
Answer: p = −2
q = 5
Call these Equation 1 and Equation 2.
Divided by (x + 1), the remainder is f(−1).
f(−1) = 2
f(−1) = (−1)⁴ + p(−1)³ + q(−1) + 6
= 1 − p − q + 6
7 − p − q = 2
−p − q = −5
Multiply both sides by −1.
p + q = 5
Equation 1: p + q = 5
Divided by (x − 3), the remainder is f(3).
f(3) = 126
f(3) = (3)⁴ + p(3)³ + q(3) + 6
= 81 + 27p + 3q + 6
87 + 27p + 3q = 126
27p + 3q = 39
Divide both sides by 3.
9p + q = 13
Equation 2: 9p + q = 13
Subtract the first equation from the second.
8p = 8
Divide both sides by 8.
p = 1
Substitute p = 1.
1 + q = 5
q = 4
Answer: p = 1
q = 4
Call these Equation 1 and Equation 2.
Divided by (x + 1), the remainder is f(−1).
f(−1) = 3
f(−1) = (−1)⁴ + p(−1)³ + q(−1) + 1
= 1 − p − q + 1
2 − p − q = 3
−p − q = 1
Multiply both sides by −1.
p + q = −1
Equation 1: p + q = −1
Divided by (x − 2), the remainder is f(2).
f(2) = 27
f(2) = (2)⁴ + p(2)³ + q(2) + 1
= 16 + 8p + 2q + 1
17 + 8p + 2q = 27
8p + 2q = 10
Divide both sides by 2.
4p + q = 5
Equation 2: 4p + q = 5
Subtract the first equation from the second.
3p = 6
Divide both sides by 3.
p = 2
Substitute p = 2.
2 + q = −1
q = −3
Answer: p = 2
q = −3
Call these Equation 1 and Equation 2.
Divided by (x − 1), the remainder is f(1).
f(1) = 7
f(1) = (1)³ + p(1)² + q(1) + 5
= 1 + p + q + 5
6 + p + q = 7
p + q = 1
Equation 1: p + q = 1
Divided by (x + 2), the remainder is f(−2).
f(−2) = 13
f(−2) = (−2)³ + p(−2)² + q(−2) + 5
= −8 + 4p − 2q + 5
−3 + 4p − 2q = 13
4p − 2q = 16
Divide both sides by 2.
2p − q = 8
Equation 2: 2p − q = 8
Add the equations.
3p = 9
Divide both sides by 3.
p = 3
Substitute p = 3.
3 + q = 1
q = −2
Answer: p = 3
q = −2
Call these Equation 1 and Equation 2.
Divided by (x + 1), the remainder is f(−1).
f(−1) = 6
f(−1) = (−1)⁴ + p(−1)³ + q(−1) + 3
= 1 − p − q + 3
4 − p − q = 6
−p − q = 2
Multiply both sides by −1.
p + q = −2
Equation 1: p + q = −2
Divided by (x − 2), the remainder is f(2).
f(2) = −9
f(2) = (2)⁴ + p(2)³ + q(2) + 3
= 16 + 8p + 2q + 3
19 + 8p + 2q = −9
8p + 2q = −28
Divide both sides by 2.
4p + q = −14
Equation 2: 4p + q = −14
Subtract the first equation from the second.
3p = −12
Divide both sides by 3.
p = −4
Substitute p = −4.
−4 + q = −2
q = 2
Answer: p = −4
q = 2
Call these Equation 1 and Equation 2.
Divided by (x − 1), the remainder is f(1).
f(1) = 1
f(1) = (1)⁴ + p(1)³ + q(1) − 2
= 1 + p + q − 2
−1 + p + q = 1
p + q = 2
Equation 1: p + q = 2
Divided by (x + 2), the remainder is f(−2).
f(−2) = −8
f(−2) = (−2)⁴ + p(−2)³ + q(−2) − 2
= 16 − 8p − 2q − 2
14 − 8p − 2q = −8
−8p − 2q = −22
Multiply both sides by −1.
8p + 2q = 22
Divide both sides by 2.
4p + q = 11
Equation 2: 4p + q = 11
Subtract the first equation from the second.
3p = 9
Divide both sides by 3.
p = 3
Substitute p = 3.
3 + q = 2
q = −1
Answer: p = 3
q = −1
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