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Answer :
To divide the polynomial [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] using long division, we'll follow these steps:
1. Set up the division: Write the dividend [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
2. Divide the leading terms:
- Divide [tex]\(-3x^5\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(-3x^2\)[/tex].
- This is the first term of our quotient.
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(-3x^2\)[/tex] resulting in [tex]\(-3x^5 - 18x^4 + 9x^3 + 15x^2\)[/tex].
- Subtract this from the original polynomial:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
= -4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the process:
- Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(-4x\)[/tex].
- Multiply the divisor by [tex]\(-4x\)[/tex], which gives [tex]\(-4x^4 - 24x^3 + 12x^2 + 20x\)[/tex].
- Subtract from the current remainder:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
= 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue with the next term:
- Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(2\)[/tex].
- Multiply the divisor by 2, resulting in [tex]\(2x^3 + 12x^2 - 6x - 10\)[/tex].
- Subtract from the remainder:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
= 4
\][/tex]
6. Conclusion:
- The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex].
- The remainder is 4.
Therefore, the result of the division is:
[tex]\[
\text{Quotient}: -3x^2 - 4x + 2
\][/tex]
[tex]\[
\text{Remainder}: 4
\][/tex]
This means that:
[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
1. Set up the division: Write the dividend [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
2. Divide the leading terms:
- Divide [tex]\(-3x^5\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(-3x^2\)[/tex].
- This is the first term of our quotient.
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(-3x^2\)[/tex] resulting in [tex]\(-3x^5 - 18x^4 + 9x^3 + 15x^2\)[/tex].
- Subtract this from the original polynomial:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
= -4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the process:
- Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(-4x\)[/tex].
- Multiply the divisor by [tex]\(-4x\)[/tex], which gives [tex]\(-4x^4 - 24x^3 + 12x^2 + 20x\)[/tex].
- Subtract from the current remainder:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
= 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue with the next term:
- Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(2\)[/tex].
- Multiply the divisor by 2, resulting in [tex]\(2x^3 + 12x^2 - 6x - 10\)[/tex].
- Subtract from the remainder:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
= 4
\][/tex]
6. Conclusion:
- The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex].
- The remainder is 4.
Therefore, the result of the division is:
[tex]\[
\text{Quotient}: -3x^2 - 4x + 2
\][/tex]
[tex]\[
\text{Remainder}: 4
\][/tex]
This means that:
[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
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