Answer :
Let's perform long division step-by-step 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].
1. Setup the division: Write the dividend [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] under the division bracket, and the divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] outside.
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\(-3x^5\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex].
- [tex]\(-3x^5 \div x^3 = -3x^2\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2.
\][/tex]
- Subtract this result from the dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2) - (-3x^5 - 18x^4 + 9x^3 + 15x^2) = -4x^4 - 22x^3 + 24x^2.
\][/tex]
4. Repeat the process:
- Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\(-4x^4 \div x^3 = -4x\)[/tex].
- Multiply the divisor by [tex]\(-4x\)[/tex]:
[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x.
\][/tex]
- Subtract from the current remainder:
[tex]\[
(-4x^4 - 22x^3 + 24x^2) - (-4x^4 - 24x^3 + 12x^2 + 20x) = 2x^3 + 12x^2 - 20x.
\][/tex]
5. Continue:
- Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\(2x^3 \div x^3 = 2\)[/tex].
- Multiply the divisor by [tex]\(2\)[/tex]:
[tex]\[
(2)(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10.
\][/tex]
- Subtract from the current remainder:
[tex]\[
(2x^3 + 12x^2 - 20x) - (2x^3 + 12x^2 - 6x - 10) = -14x + 10.
\][/tex]
6. Final Remainder: The degree of the remainder [tex]\(-14x + 10\)[/tex] is less than the degree of the divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex], so we are done.
The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(-14x + 10\)[/tex].
Therefore, the result of the division is:
[tex]\[
\boxed{-3x^2 - 4x + 2} \quad \text{with a remainder of } \boxed{-14x + 10}
\][/tex]
1. Setup the division: Write the dividend [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] under the division bracket, and the divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] outside.
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\(-3x^5\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex].
- [tex]\(-3x^5 \div x^3 = -3x^2\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2.
\][/tex]
- Subtract this result from the dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2) - (-3x^5 - 18x^4 + 9x^3 + 15x^2) = -4x^4 - 22x^3 + 24x^2.
\][/tex]
4. Repeat the process:
- Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\(-4x^4 \div x^3 = -4x\)[/tex].
- Multiply the divisor by [tex]\(-4x\)[/tex]:
[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x.
\][/tex]
- Subtract from the current remainder:
[tex]\[
(-4x^4 - 22x^3 + 24x^2) - (-4x^4 - 24x^3 + 12x^2 + 20x) = 2x^3 + 12x^2 - 20x.
\][/tex]
5. Continue:
- Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\(2x^3 \div x^3 = 2\)[/tex].
- Multiply the divisor by [tex]\(2\)[/tex]:
[tex]\[
(2)(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10.
\][/tex]
- Subtract from the current remainder:
[tex]\[
(2x^3 + 12x^2 - 20x) - (2x^3 + 12x^2 - 6x - 10) = -14x + 10.
\][/tex]
6. Final Remainder: The degree of the remainder [tex]\(-14x + 10\)[/tex] is less than the degree of the divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex], so we are done.
The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(-14x + 10\)[/tex].
Therefore, the result of the division is:
[tex]\[
\boxed{-3x^2 - 4x + 2} \quad \text{with a remainder of } \boxed{-14x + 10}
\][/tex]
