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Answer :
To solve this problem using synthetic division, follow these steps:
1. Identify the Coefficients: Write down the coefficients of the polynomial [tex]\(2x^4 - 12x^3 + 23x^2 - 42x + 56\)[/tex]. These are: [tex]\(2, -12, 23, -42, 56\)[/tex].
2. Determine the Root: Since we are dividing by [tex]\(x - 4\)[/tex], the root is [tex]\(4\)[/tex].
3. Set Up Synthetic Division: Begin the synthetic division by writing the root (4) to the left, and the coefficients next to it in a row.
4. Start the Process:
- Bring down the first coefficient (2) as it is.
5. Multiply and Add:
- Multiply the root (4) by the last number you brought down (2). [tex]\(4 \times 2 = 8\)[/tex].
- Add this result to the next coefficient: [tex]\(-12 + 8 = -4\)[/tex].
- Write [tex]\(-4\)[/tex] below the line.
6. Repeat the Multiplication and Addition:
- Multiply the root (4) by [tex]\(-4\)[/tex]. [tex]\(4 \times -4 = -16\)[/tex].
- Add this result to the next coefficient: [tex]\(23 + (-16) = 7\)[/tex].
- Write [tex]\(7\)[/tex] below the line.
7. Continue the Process:
- Multiply the root (4) by [tex]\(7\)[/tex]. [tex]\(4 \times 7 = 28\)[/tex].
- Add this to the next coefficient: [tex]\(-42 + 28 = -14\)[/tex].
- Write [tex]\(-14\)[/tex] below the line.
8. Final Calculation:
- Multiply the root (4) by [tex]\(-14\)[/tex]. [tex]\(4 \times -14 = -56\)[/tex].
- Add this to the last coefficient: [tex]\(56 + (-56) = 0\)[/tex].
9. Interpret the Results:
- The numbers on the bottom row, except the last one, are the coefficients of the quotient: [tex]\(2, -4, 7, -14\)[/tex].
- The last number, which is 0, is the remainder.
Therefore, after dividing [tex]\(2x^4 - 12x^3 + 23x^2 - 42x + 56\)[/tex] by [tex]\(x - 4\)[/tex], the quotient is [tex]\(2x^3 - 4x^2 + 7x - 14\)[/tex] with no remainder.
1. Identify the Coefficients: Write down the coefficients of the polynomial [tex]\(2x^4 - 12x^3 + 23x^2 - 42x + 56\)[/tex]. These are: [tex]\(2, -12, 23, -42, 56\)[/tex].
2. Determine the Root: Since we are dividing by [tex]\(x - 4\)[/tex], the root is [tex]\(4\)[/tex].
3. Set Up Synthetic Division: Begin the synthetic division by writing the root (4) to the left, and the coefficients next to it in a row.
4. Start the Process:
- Bring down the first coefficient (2) as it is.
5. Multiply and Add:
- Multiply the root (4) by the last number you brought down (2). [tex]\(4 \times 2 = 8\)[/tex].
- Add this result to the next coefficient: [tex]\(-12 + 8 = -4\)[/tex].
- Write [tex]\(-4\)[/tex] below the line.
6. Repeat the Multiplication and Addition:
- Multiply the root (4) by [tex]\(-4\)[/tex]. [tex]\(4 \times -4 = -16\)[/tex].
- Add this result to the next coefficient: [tex]\(23 + (-16) = 7\)[/tex].
- Write [tex]\(7\)[/tex] below the line.
7. Continue the Process:
- Multiply the root (4) by [tex]\(7\)[/tex]. [tex]\(4 \times 7 = 28\)[/tex].
- Add this to the next coefficient: [tex]\(-42 + 28 = -14\)[/tex].
- Write [tex]\(-14\)[/tex] below the line.
8. Final Calculation:
- Multiply the root (4) by [tex]\(-14\)[/tex]. [tex]\(4 \times -14 = -56\)[/tex].
- Add this to the last coefficient: [tex]\(56 + (-56) = 0\)[/tex].
9. Interpret the Results:
- The numbers on the bottom row, except the last one, are the coefficients of the quotient: [tex]\(2, -4, 7, -14\)[/tex].
- The last number, which is 0, is the remainder.
Therefore, after dividing [tex]\(2x^4 - 12x^3 + 23x^2 - 42x + 56\)[/tex] by [tex]\(x - 4\)[/tex], the quotient is [tex]\(2x^3 - 4x^2 + 7x - 14\)[/tex] with no remainder.
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