Answer :
To solve the problem of dividing the polynomial [tex]\(4x^4 - 27x^3 + 19x^2 - 7x + 6\)[/tex] by [tex]\(x - 6\)[/tex], we can use the method of synthetic division. Here is a step-by-step explanation of the process:
1. Identify the Dividend and Divisor:
- The dividend is the polynomial [tex]\(4x^4 - 27x^3 + 19x^2 - 7x + 6\)[/tex].
- The divisor is [tex]\(x - 6\)[/tex]. The zero of the divisor is [tex]\(6\)[/tex].
2. Set Up Synthetic Division:
- Write down the coefficients of the dividend polynomial: [tex]\(4, -27, 19, -7, 6\)[/tex].
- Use the zero of the divisor, [tex]\(6\)[/tex], for the synthetic division process.
3. Perform Synthetic Division:
- Bring down the first coefficient, [tex]\(4\)[/tex], as it is.
- Multiply [tex]\(4\)[/tex] by [tex]\(6\)[/tex] (the divisor value) and add to the next coefficient [tex]\(-27\)[/tex]:
[tex]\(4 \times 6 = 24\)[/tex]; [tex]\(-27 + 24 = -3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(6\)[/tex] and add to the next coefficient [tex]\(19\)[/tex]:
[tex]\(-3 \times 6 = -18\)[/tex]; [tex]\(19 + (-18) = 1\)[/tex].
- Multiply [tex]\(1\)[/tex] by [tex]\(6\)[/tex] and add to the next coefficient [tex]\(-7\)[/tex]:
[tex]\(1 \times 6 = 6\)[/tex]; [tex]\(-7 + 6 = -1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\(6\)[/tex] and add to the last coefficient [tex]\(6\)[/tex]:
[tex]\(-1 \times 6 = -6\)[/tex]; [tex]\(6 + (-6) = 0\)[/tex].
4. Write Down the Results:
- The coefficients of the quotient polynomial are [tex]\(4, -3, 1, -1\)[/tex].
- The remainder is [tex]\(0\)[/tex], indicating an exact division.
5. Form the Quotient Polynomial:
- The quotient polynomial is formed using the coefficients:
[tex]\(4x^3 - 3x^2 + x - 1\)[/tex].
In conclusion, the division of [tex]\(4x^4 - 27x^3 + 19x^2 - 7x + 6\)[/tex] by [tex]\(x - 6\)[/tex] results in the quotient [tex]\(4x^3 - 3x^2 + x - 1\)[/tex] with a remainder of [tex]\(0\)[/tex].
1. Identify the Dividend and Divisor:
- The dividend is the polynomial [tex]\(4x^4 - 27x^3 + 19x^2 - 7x + 6\)[/tex].
- The divisor is [tex]\(x - 6\)[/tex]. The zero of the divisor is [tex]\(6\)[/tex].
2. Set Up Synthetic Division:
- Write down the coefficients of the dividend polynomial: [tex]\(4, -27, 19, -7, 6\)[/tex].
- Use the zero of the divisor, [tex]\(6\)[/tex], for the synthetic division process.
3. Perform Synthetic Division:
- Bring down the first coefficient, [tex]\(4\)[/tex], as it is.
- Multiply [tex]\(4\)[/tex] by [tex]\(6\)[/tex] (the divisor value) and add to the next coefficient [tex]\(-27\)[/tex]:
[tex]\(4 \times 6 = 24\)[/tex]; [tex]\(-27 + 24 = -3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(6\)[/tex] and add to the next coefficient [tex]\(19\)[/tex]:
[tex]\(-3 \times 6 = -18\)[/tex]; [tex]\(19 + (-18) = 1\)[/tex].
- Multiply [tex]\(1\)[/tex] by [tex]\(6\)[/tex] and add to the next coefficient [tex]\(-7\)[/tex]:
[tex]\(1 \times 6 = 6\)[/tex]; [tex]\(-7 + 6 = -1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\(6\)[/tex] and add to the last coefficient [tex]\(6\)[/tex]:
[tex]\(-1 \times 6 = -6\)[/tex]; [tex]\(6 + (-6) = 0\)[/tex].
4. Write Down the Results:
- The coefficients of the quotient polynomial are [tex]\(4, -3, 1, -1\)[/tex].
- The remainder is [tex]\(0\)[/tex], indicating an exact division.
5. Form the Quotient Polynomial:
- The quotient polynomial is formed using the coefficients:
[tex]\(4x^3 - 3x^2 + x - 1\)[/tex].
In conclusion, the division of [tex]\(4x^4 - 27x^3 + 19x^2 - 7x + 6\)[/tex] by [tex]\(x - 6\)[/tex] results in the quotient [tex]\(4x^3 - 3x^2 + x - 1\)[/tex] with a remainder of [tex]\(0\)[/tex].
