Answer :
To solve the problem of dividing the polynomial [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex] by [tex]\(x + 3\)[/tex] using synthetic division, follow these steps:
1. Identify the root of the divisor: The divisor is [tex]\(x + 3\)[/tex]. To use synthetic division, we need to find the root, which is the value that makes the divisor zero. For [tex]\(x + 3 = 0\)[/tex], the root is [tex]\(x = -3\)[/tex].
2. Write down the coefficients: List the coefficients of the polynomial [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex]. These coefficients are [1, -5, -23, 5, 6].
3. Set up synthetic division:
- Write the root [tex]\(-3\)[/tex] to the left.
- Write the coefficients [1, -5, -23, 5, 6] in a row next to the root.
4. Start the division process:
- Bring down the first coefficient (1) to the bottom row as it is.
5. Perform the synthetic division steps:
- Multiply the root (-3) by the number on the bottom row (starting with the 1 that was brought down) and write the result under the next coefficient.
- Add this result to the next coefficient:
- [tex]\(1 \times (-3) = -3\)[/tex]; [tex]\(-5 + (-3) = -8\)[/tex]
- [tex]\(-8 \times (-3) = 24\)[/tex]; [tex]\(-23 + 24 = 1\)[/tex]
- [tex]\(1 \times (-3) = -3\)[/tex]; [tex]\(5 + (-3) = 2\)[/tex]
- [tex]\(2 \times (-3) = -6\)[/tex]; [tex]\(6 + (-6) = 0\)[/tex]
6. Interpret the results:
- The numbers on the bottom row, except the last one, are the coefficients of the quotient polynomial. Here, these are [1, -8, 1, 2].
- The last number is the remainder, which is 0 in this case.
7. Write the final result:
- The quotient polynomial, therefore, is [tex]\(x^3 - 8x^2 + x + 2\)[/tex].
- Since the remainder is 0, it means that [tex]\(x + 3\)[/tex] is a factor of the original polynomial.
Therefore, the result of dividing [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex] by [tex]\(x + 3\)[/tex] is [tex]\(x^3 - 8x^2 + x + 2\)[/tex] with a remainder of 0.
1. Identify the root of the divisor: The divisor is [tex]\(x + 3\)[/tex]. To use synthetic division, we need to find the root, which is the value that makes the divisor zero. For [tex]\(x + 3 = 0\)[/tex], the root is [tex]\(x = -3\)[/tex].
2. Write down the coefficients: List the coefficients of the polynomial [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex]. These coefficients are [1, -5, -23, 5, 6].
3. Set up synthetic division:
- Write the root [tex]\(-3\)[/tex] to the left.
- Write the coefficients [1, -5, -23, 5, 6] in a row next to the root.
4. Start the division process:
- Bring down the first coefficient (1) to the bottom row as it is.
5. Perform the synthetic division steps:
- Multiply the root (-3) by the number on the bottom row (starting with the 1 that was brought down) and write the result under the next coefficient.
- Add this result to the next coefficient:
- [tex]\(1 \times (-3) = -3\)[/tex]; [tex]\(-5 + (-3) = -8\)[/tex]
- [tex]\(-8 \times (-3) = 24\)[/tex]; [tex]\(-23 + 24 = 1\)[/tex]
- [tex]\(1 \times (-3) = -3\)[/tex]; [tex]\(5 + (-3) = 2\)[/tex]
- [tex]\(2 \times (-3) = -6\)[/tex]; [tex]\(6 + (-6) = 0\)[/tex]
6. Interpret the results:
- The numbers on the bottom row, except the last one, are the coefficients of the quotient polynomial. Here, these are [1, -8, 1, 2].
- The last number is the remainder, which is 0 in this case.
7. Write the final result:
- The quotient polynomial, therefore, is [tex]\(x^3 - 8x^2 + x + 2\)[/tex].
- Since the remainder is 0, it means that [tex]\(x + 3\)[/tex] is a factor of the original polynomial.
Therefore, the result of dividing [tex]\(x^4 - 5x^3 - 23x^2 + 5x + 6\)[/tex] by [tex]\(x + 3\)[/tex] is [tex]\(x^3 - 8x^2 + x + 2\)[/tex] with a remainder of 0.
