Remainder Theorem

Show your work for each problem.

1.
According to the Remainder Theorem, when is divided by , what is the remainder?
1. a)
2. b)
3. c)
4. d)
Answer:
. The remainder is , which means is a factor!
2.
Find the remainder when is divided by .
Answer: 0
. Since the remainder is , is a factor of .
3.
When dividing by , what value should you substitute into to find the remainder?
1. a)
2. b)
3. c)
4. d)
Answer:
, so . Substitute to find the remainder.
4.
Find the remainder when is divided by .
Answer: 0
. The polynomial is divisible by .
5.
Find the remainder when is divided by .
Answer: 4
1. What value of should you substitute? (divisor is ) 1
2. What is ? 2
3. What is ? 5
4. Calculate 4
6.
Find the remainder when is divided by .
Answer: 0
. Since the remainder is , is a factor of .
7.
What is the remainder when is divided by ?
1. a)
2. b)
3. c)
4. d)
Answer:
8.
Find the remainder when is divided by .
Answer: -10
1. What value do you substitute for ? -1
2. What is ? -3
3. What is ? -1
4. What is ? -2
5. Calculate -10
9.
If is a factor of , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
, so , therefore .
10.
Is a factor of ? Use the Remainder Theorem to verify.
Answer: 0
1. What value do you substitute for ? -3
2. What is ? -27
3. What is ? 18
4. What is ? 15
5. Calculate 0
11.
Find the remainder when is divided by .
Answer: 5
. The remainder is .
12.
If has remainder when divided by both and , find .
Answer: 4
1. From : . Simplify to get 5
2. From : . Simplify to get 14
3. Simplify equation 2: 7
4. Add equations: . So 12
5. Solve for : 4
13.
Find the remainder when is divided by .
Answer: 3
. No matter the exponent, !
14.
What is the remainder when is divided by ?
1. a)
2. b)
3. c)
4. d)
Answer:
because 99 is odd. An odd power of equals .
15.
If has remainder when divided by , find .
Answer: -5
1. By the Remainder Theorem, 2
2. What is ? 3
3. Set up equation: . What is ? 2
4. Solve for : -5
16.
The polynomial is divisible by . What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
, so .
17.
Find the remainder when is divided by .
Answer: 1
. The alternating signs give remainder .
18.
Find the remainder when is divided by .
Answer: 25
1. What is ? 54
2. What is ? -45
3. What is ? 9
4. Calculate 25