Dividing Polynomials

Show your work for each problem.

1.
Simplify:
1. a)
2. b)
3. c)
4. d)
Answer:
2.
Simplify: . Enter the simplified form (e.g., 3x^2).
Answer: 3x^2
3.
Simplify:
Answer: 4x + 2
1. Divide the first term: = ? 4x
2. Divide the second term: = ? 2
3. Combine the results: 4x + 2
4.
In synthetic division, what value of do you use to divide by ?
1. a)
2. b)
3. c)
4. d)
Answer:
For , we have . The sign in synthetic division is always opposite to what appears in the divisor.
5.
Use polynomial long division to divide by .
Answer: x + 4
1. Divide by : x
2. Multiply by and subtract from . What remains? 4x
3. Bring down . Now divide by : 4
4. Multiply by and subtract from . What is the remainder? 0
6.
Using the Remainder Theorem, what is the remainder when is divided by ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Since the remainder is , is a factor of .
7.
Divide by . Enter the quotient.
Answer: x - 3
. Dividing by gives with no remainder.
8.
Use synthetic division to divide by .
Answer: x^2 + 4x + 3
1. What value of do you use? 2
2. Write the coefficients: . Bring down the 1. Multiply by 2 and add to 2: 4
3. Multiply 4 by 2, add to -5: 3
4. Multiply 3 by 2, add to -6. What is the remainder? 0
9.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
10.
Divide by using long division.
Answer: 2x - 1
1. Divide by : 2x
2. Multiply by and subtract from . What remains? -x
3. Bring down . Now divide by : -1
4. Multiply by and subtract from . What is the remainder? 0
11.
Use synthetic division to find the remainder when is divided by .
Answer: 0
Using : 3 | 1 -4 1 6 gives 1, -1, -2, 0. The remainder is 0, so is a factor.
12.
Divide by using synthetic division. Remember to include zero coefficients!
Answer: x^2 + 2x + 4
1. Write the polynomial with all terms. What are the coefficients? 1, 0, 0, -8
2. Set up synthetic division with . After the first operation: 2
3. Continue: 4
4. Final step: . This is the remainder. 0
13.
What is the quotient when is divided by ?
1. a) (remainder -8)
2. b) (remainder 0)
3. c) (remainder 0)
4. d) (remainder 8)
Answer: (remainder -8)
Using synthetic division with : We write 3, 5, -2, -8 (coefficients). Bring down 3. Then , add to 5 gives -1. Then , add to -2 gives 0. Then , add to -8 gives -8. The quotient is with remainder . Verify: ✓
14.
Divide by . Enter the quotient polynomial.
Answer: x^3 + 2x^2 + 4x + 8
Using synthetic division: 2 | 1 0 0 0 -16 gives 1, 2, 4, 8, 0. Quotient: .
15.
Divide by using polynomial long division.
Answer: 3x^2 + x - 5
1. Divide by : 3x^2
2. Multiply by : . Subtract from . What remains? 2x^2
3. Bring down . Divide by : x
4. Multiply by : . Subtract from . What remains? -10x
5. Bring down . Divide by : -5
6. Multiply by : . Subtract from . What is the remainder? 1