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Answer :
Certainly! We can solve this problem by performing polynomial long division of the given polynomials.
Step-by-step Polynomial Long Division:
We need to divide the numerator [tex]\( -3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6 \)[/tex] by the denominator [tex]\( x^3 + 6x^2 - 3x - 5 \)[/tex].
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This becomes the first term of our quotient.
2. Multiply the entire divisor by this term:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
3. Subtract the result from the original numerator:
[tex]\[
(-3x^5 - 22x^4 - 13x^3) - (-3x^5 - 18x^4 + 9x^3) = 0x^5 - 4x^4 - 22x^3
\][/tex]
Bring down the next term from the original numerator:
[tex]\[
-4x^4 - 22x^3 + 39x^2
\][/tex]
4. Repeat the process:
- Divide the first new term by the first term of the divisor:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
- Multiply the entire divisor by this term:
[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
- Subtract:
[tex]\[
(-4x^4 - 22x^3 + 39x^2) - (-4x^4 - 24x^3 + 12x^2) = 0x^4 + 2x^3 + 27x^2
\][/tex]
Bring down the next term:
[tex]\[
2x^3 + 27x^2 + 14x
\][/tex]
5. Repeat again:
- Divide:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
- Multiply:
[tex]\[
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
- Subtract:
[tex]\[
(2x^3 + 27x^2 + 14x) - (2x^3 + 12x^2 - 6x) = 0x^3 + 15x^2 + 20x
\][/tex]
Bring down the final term:
[tex]\[
15x^2 + 20x - 6
\][/tex]
6. Final step: Check remainder
At this stage, our result from subtraction is [tex]\( 15x^2 + 20x - 6 \)[/tex], which is lower degree than the divisor, so it becomes the remainder.
Conclusion:
The quotient is:
[tex]\[
-3x^2 - 4x + 2
\][/tex]
The remainder is:
[tex]\[
15x^2 + 20x - 6
\][/tex]
Thus, the division of the polynomials is:
[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{15x^2 + 20x - 6}{x^3 + 6x^2 - 3x - 5}
\][/tex]
Step-by-step Polynomial Long Division:
We need to divide the numerator [tex]\( -3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6 \)[/tex] by the denominator [tex]\( x^3 + 6x^2 - 3x - 5 \)[/tex].
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This becomes the first term of our quotient.
2. Multiply the entire divisor by this term:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
3. Subtract the result from the original numerator:
[tex]\[
(-3x^5 - 22x^4 - 13x^3) - (-3x^5 - 18x^4 + 9x^3) = 0x^5 - 4x^4 - 22x^3
\][/tex]
Bring down the next term from the original numerator:
[tex]\[
-4x^4 - 22x^3 + 39x^2
\][/tex]
4. Repeat the process:
- Divide the first new term by the first term of the divisor:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
- Multiply the entire divisor by this term:
[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
- Subtract:
[tex]\[
(-4x^4 - 22x^3 + 39x^2) - (-4x^4 - 24x^3 + 12x^2) = 0x^4 + 2x^3 + 27x^2
\][/tex]
Bring down the next term:
[tex]\[
2x^3 + 27x^2 + 14x
\][/tex]
5. Repeat again:
- Divide:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
- Multiply:
[tex]\[
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
- Subtract:
[tex]\[
(2x^3 + 27x^2 + 14x) - (2x^3 + 12x^2 - 6x) = 0x^3 + 15x^2 + 20x
\][/tex]
Bring down the final term:
[tex]\[
15x^2 + 20x - 6
\][/tex]
6. Final step: Check remainder
At this stage, our result from subtraction is [tex]\( 15x^2 + 20x - 6 \)[/tex], which is lower degree than the divisor, so it becomes the remainder.
Conclusion:
The quotient is:
[tex]\[
-3x^2 - 4x + 2
\][/tex]
The remainder is:
[tex]\[
15x^2 + 20x - 6
\][/tex]
Thus, the division of the polynomials is:
[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{15x^2 + 20x - 6}{x^3 + 6x^2 - 3x - 5}
\][/tex]
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