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Answer :
Sure! Let's go through the synthetic division process for both problems step-by-step:
### 1. Dividing [tex]\((2x^4 + 4x^3 - 2x^2 + 8x - 4) \div (x + 3)\)[/tex]:
Step 1: Identify the divisor.
The divisor is [tex]\(x + 3\)[/tex], so the root we use for synthetic division is [tex]\(-3\)[/tex].
Step 2: Set up the synthetic division.
Write down the coefficients of the polynomial: [tex]\(2, 4, -2, 8, -4\)[/tex].
Place [tex]\(-3\)[/tex] on the left side.
Step 3: Start the synthetic division process.
1. Bring down the first coefficient [tex]\(2\)[/tex].
2. Multiply [tex]\(-3\)[/tex] by [tex]\(2\)[/tex] and write it under the next coefficient [tex]\(4\)[/tex].
3. Add to get [tex]\(4 + (-6) = -2\)[/tex].
4. Multiply [tex]\(-3\)[/tex] by [tex]\(-2\)[/tex] to get [tex]\(6\)[/tex] and write under the next coefficient [tex]\(-2\)[/tex].
5. Add to get [tex]\(-2 + 6 = 4\)[/tex].
6. Multiply [tex]\(-3\)[/tex] by [tex]\(4\)[/tex] to get [tex]\(-12\)[/tex] and write under the next coefficient [tex]\(8\)[/tex].
7. Add to get [tex]\(8 + (-12) = -4\)[/tex].
8. Multiply [tex]\(-3\)[/tex] by [tex]\(-4\)[/tex] to get [tex]\(12\)[/tex] and write under the next coefficient [tex]\(-4\)[/tex].
9. Add to get [tex]\(-4 + 12 = 8\)[/tex].
Result:
The quotient is [tex]\(2x^3 - 2x^2 + 4x - 4\)[/tex], and the remainder is [tex]\(8\)[/tex].
### 2. Dividing [tex]\((12x^5 + 10x^4 - 18x^3 - 12x^2 - 8) \div (2x - 3)\)[/tex]:
Step 1: Identify the divisor.
The divisor is [tex]\(2x - 3\)[/tex], so the root for synthetic division is [tex]\(\frac{3}{2}\)[/tex].
Step 2: Set up the synthetic division.
Write down the coefficients of the polynomial: [tex]\(12, 10, -18, -12, 0, -8\)[/tex].
Place [tex]\(\frac{3}{2}\)[/tex] on the left side.
Step 3: Start the synthetic division process.
1. Bring down the first coefficient [tex]\(12\)[/tex].
2. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(12\)[/tex] to get [tex]\(18\)[/tex] and write it under the next coefficient [tex]\(10\)[/tex].
3. Add to get [tex]\(10 + 18 = 28\)[/tex].
4. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(28\)[/tex] to get [tex]\(42\)[/tex] and write it under the next coefficient [tex]\(-18\)[/tex].
5. Add to get [tex]\(-18 + 42 = 24\)[/tex].
6. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(24\)[/tex] to get [tex]\(36\)[/tex] and write it under the next coefficient [tex]\(-12\)[/tex].
7. Add to get [tex]\(-12 + 36 = 24\)[/tex].
8. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(24\)[/tex] to get [tex]\(36\)[/tex] and write it under the next coefficient [tex]\(0\)[/tex].
9. Add to get [tex]\(0 + 36 = 36\)[/tex].
10. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(36\)[/tex] to get [tex]\(54\)[/tex] and write it under the next coefficient [tex]\(-8\)[/tex].
11. Add to get [tex]\(-8 + 54 = 46\)[/tex].
Result:
The quotient is [tex]\(12x^4 + 28x^3 + 24x^2 + 24x + 36\)[/tex], and the remainder is [tex]\(46\)[/tex].
That's the complete step-by-step solution for both divisions using synthetic division! Let me know if you have any more questions.
### 1. Dividing [tex]\((2x^4 + 4x^3 - 2x^2 + 8x - 4) \div (x + 3)\)[/tex]:
Step 1: Identify the divisor.
The divisor is [tex]\(x + 3\)[/tex], so the root we use for synthetic division is [tex]\(-3\)[/tex].
Step 2: Set up the synthetic division.
Write down the coefficients of the polynomial: [tex]\(2, 4, -2, 8, -4\)[/tex].
Place [tex]\(-3\)[/tex] on the left side.
Step 3: Start the synthetic division process.
1. Bring down the first coefficient [tex]\(2\)[/tex].
2. Multiply [tex]\(-3\)[/tex] by [tex]\(2\)[/tex] and write it under the next coefficient [tex]\(4\)[/tex].
3. Add to get [tex]\(4 + (-6) = -2\)[/tex].
4. Multiply [tex]\(-3\)[/tex] by [tex]\(-2\)[/tex] to get [tex]\(6\)[/tex] and write under the next coefficient [tex]\(-2\)[/tex].
5. Add to get [tex]\(-2 + 6 = 4\)[/tex].
6. Multiply [tex]\(-3\)[/tex] by [tex]\(4\)[/tex] to get [tex]\(-12\)[/tex] and write under the next coefficient [tex]\(8\)[/tex].
7. Add to get [tex]\(8 + (-12) = -4\)[/tex].
8. Multiply [tex]\(-3\)[/tex] by [tex]\(-4\)[/tex] to get [tex]\(12\)[/tex] and write under the next coefficient [tex]\(-4\)[/tex].
9. Add to get [tex]\(-4 + 12 = 8\)[/tex].
Result:
The quotient is [tex]\(2x^3 - 2x^2 + 4x - 4\)[/tex], and the remainder is [tex]\(8\)[/tex].
### 2. Dividing [tex]\((12x^5 + 10x^4 - 18x^3 - 12x^2 - 8) \div (2x - 3)\)[/tex]:
Step 1: Identify the divisor.
The divisor is [tex]\(2x - 3\)[/tex], so the root for synthetic division is [tex]\(\frac{3}{2}\)[/tex].
Step 2: Set up the synthetic division.
Write down the coefficients of the polynomial: [tex]\(12, 10, -18, -12, 0, -8\)[/tex].
Place [tex]\(\frac{3}{2}\)[/tex] on the left side.
Step 3: Start the synthetic division process.
1. Bring down the first coefficient [tex]\(12\)[/tex].
2. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(12\)[/tex] to get [tex]\(18\)[/tex] and write it under the next coefficient [tex]\(10\)[/tex].
3. Add to get [tex]\(10 + 18 = 28\)[/tex].
4. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(28\)[/tex] to get [tex]\(42\)[/tex] and write it under the next coefficient [tex]\(-18\)[/tex].
5. Add to get [tex]\(-18 + 42 = 24\)[/tex].
6. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(24\)[/tex] to get [tex]\(36\)[/tex] and write it under the next coefficient [tex]\(-12\)[/tex].
7. Add to get [tex]\(-12 + 36 = 24\)[/tex].
8. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(24\)[/tex] to get [tex]\(36\)[/tex] and write it under the next coefficient [tex]\(0\)[/tex].
9. Add to get [tex]\(0 + 36 = 36\)[/tex].
10. Multiply [tex]\(\frac{3}{2}\)[/tex] by [tex]\(36\)[/tex] to get [tex]\(54\)[/tex] and write it under the next coefficient [tex]\(-8\)[/tex].
11. Add to get [tex]\(-8 + 54 = 46\)[/tex].
Result:
The quotient is [tex]\(12x^4 + 28x^3 + 24x^2 + 24x + 36\)[/tex], and the remainder is [tex]\(46\)[/tex].
That's the complete step-by-step solution for both divisions using synthetic division! Let me know if you have any more questions.
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