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Answer :
Certainly! Let's find the quotient when [tex]\(-12x^4 + 10x^3 - 23x^2 - 17x + 8\)[/tex] is divided by [tex]\(-3x + 1\)[/tex] using polynomial long division.
### Polynomial Long Division
1. Setup the Division:
- Dividend: [tex]\(-12x^4 + 10x^3 - 23x^2 - 17x + 8\)[/tex]
- Divisor: [tex]\(-3x + 1\)[/tex]
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(-12x^4\)[/tex], by the leading term of the divisor, [tex]\(-3x\)[/tex].
- [tex]\(\frac{-12x^4}{-3x} = 4x^3\)[/tex]
- The first term of the quotient is [tex]\(4x^3\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\((-3x + 1)\)[/tex] by [tex]\(4x^3\)[/tex]:
[tex]\(-12x^4 + 4x^3\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(-12x^4 + 10x^3 - 23x^2 - 17x + 8) - (-12x^4 + 4x^3) = 6x^3 - 23x^2 - 17x + 8
\][/tex]
4. Repeat the Process:
- Divide [tex]\(6x^3\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\(\frac{6x^3}{-3x} = -2x^2\)[/tex]
- Multiply the divisor [tex]\((-3x + 1)\)[/tex] by [tex]\(-2x^2\)[/tex]:
[tex]\(-6x^3 + 2x^2\)[/tex].
- Subtract:
[tex]\[
(6x^3 - 23x^2 - 17x + 8) - (-6x^3 + 2x^2) = -21x^2 - 17x + 8
\][/tex]
5. Continue with Next Term:
- Divide [tex]\(-21x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\(\frac{-21x^2}{-3x} = 7x\)[/tex]
- Multiply the divisor [tex]\((-3x + 1)\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\(-21x^2 + 7x\)[/tex].
- Subtract:
[tex]\[
(-21x^2 - 17x + 8) - (-21x^2 + 7x) = -24x + 8
\][/tex]
6. Final Step:
- Divide [tex]\(-24x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\(\frac{-24x}{-3x} = 8\)[/tex]
- Multiply the divisor [tex]\((-3x + 1)\)[/tex] by 8:
[tex]\(-24x + 8\)[/tex].
- Subtract:
[tex]\[
(-24x + 8) - (-24x + 8) = 0
\][/tex]
The quotient resulting from this division is:
[tex]\[ 4x^3 - 2x^2 + 7x + 8 \][/tex]
And there is no remainder, which completes the division process.
### Polynomial Long Division
1. Setup the Division:
- Dividend: [tex]\(-12x^4 + 10x^3 - 23x^2 - 17x + 8\)[/tex]
- Divisor: [tex]\(-3x + 1\)[/tex]
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(-12x^4\)[/tex], by the leading term of the divisor, [tex]\(-3x\)[/tex].
- [tex]\(\frac{-12x^4}{-3x} = 4x^3\)[/tex]
- The first term of the quotient is [tex]\(4x^3\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\((-3x + 1)\)[/tex] by [tex]\(4x^3\)[/tex]:
[tex]\(-12x^4 + 4x^3\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(-12x^4 + 10x^3 - 23x^2 - 17x + 8) - (-12x^4 + 4x^3) = 6x^3 - 23x^2 - 17x + 8
\][/tex]
4. Repeat the Process:
- Divide [tex]\(6x^3\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\(\frac{6x^3}{-3x} = -2x^2\)[/tex]
- Multiply the divisor [tex]\((-3x + 1)\)[/tex] by [tex]\(-2x^2\)[/tex]:
[tex]\(-6x^3 + 2x^2\)[/tex].
- Subtract:
[tex]\[
(6x^3 - 23x^2 - 17x + 8) - (-6x^3 + 2x^2) = -21x^2 - 17x + 8
\][/tex]
5. Continue with Next Term:
- Divide [tex]\(-21x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\(\frac{-21x^2}{-3x} = 7x\)[/tex]
- Multiply the divisor [tex]\((-3x + 1)\)[/tex] by [tex]\(7x\)[/tex]:
[tex]\(-21x^2 + 7x\)[/tex].
- Subtract:
[tex]\[
(-21x^2 - 17x + 8) - (-21x^2 + 7x) = -24x + 8
\][/tex]
6. Final Step:
- Divide [tex]\(-24x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\(\frac{-24x}{-3x} = 8\)[/tex]
- Multiply the divisor [tex]\((-3x + 1)\)[/tex] by 8:
[tex]\(-24x + 8\)[/tex].
- Subtract:
[tex]\[
(-24x + 8) - (-24x + 8) = 0
\][/tex]
The quotient resulting from this division is:
[tex]\[ 4x^3 - 2x^2 + 7x + 8 \][/tex]
And there is no remainder, which completes the division process.
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