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Answer :
To solve the problem of dividing the polynomial [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by the polynomial [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex], we can perform polynomial long division. Here is a step-by-step breakdown of the process:
1. Set up the division: Place the dividend, [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex], under the division symbol, and the divisor, [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex], outside.
2. Determine the first term of the quotient: Divide the leading term of the dividend, [tex]\(-3x^5\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This is the first term of the quotient.
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(-3x^2\)[/tex] and subtract the result from the dividend.
[tex]\[
(-3x^2) \times (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtracting 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 the new leading term, [tex]\(-4x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Multiply the divisor by [tex]\(-4x\)[/tex]:
[tex]\[
(-4x) \times (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x) = 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Proceed further: Divide the new leading term, [tex]\(2x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Multiply the divisor by [tex]\(2\)[/tex]:
[tex]\[
2 \times (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10) = 4
\][/tex]
6. Conclusion: The remainder is [tex]\(4\)[/tex], and the quotient is [tex]\(-3x^2 - 4x + 2\)[/tex].
Thus, the result of the division is:
[tex]\[
-3x^2 - 4x + 2 \quad \text{with a remainder of} \quad 4
\][/tex]
1. Set up the division: Place the dividend, [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex], under the division symbol, and the divisor, [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex], outside.
2. Determine the first term of the quotient: Divide the leading term of the dividend, [tex]\(-3x^5\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This is the first term of the quotient.
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(-3x^2\)[/tex] and subtract the result from the dividend.
[tex]\[
(-3x^2) \times (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtracting 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 the new leading term, [tex]\(-4x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Multiply the divisor by [tex]\(-4x\)[/tex]:
[tex]\[
(-4x) \times (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x) = 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Proceed further: Divide the new leading term, [tex]\(2x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Multiply the divisor by [tex]\(2\)[/tex]:
[tex]\[
2 \times (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10) = 4
\][/tex]
6. Conclusion: The remainder is [tex]\(4\)[/tex], and the quotient is [tex]\(-3x^2 - 4x + 2\)[/tex].
Thus, the result of the division is:
[tex]\[
-3x^2 - 4x + 2 \quad \text{with a remainder of} \quad 4
\][/tex]
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