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Answer :
Sure! Let's perform long division for the expression [tex]\(\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5}\)[/tex].
Step 1: Set up the long division.
The dividend is [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor is [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
Step 2: Divide the first term of the dividend by the first term of the divisor.
Divide [tex]\(-3x^5\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(-3x^2\)[/tex].
Step 3: Multiply the entire divisor by this quotient term.
Multiply [tex]\(-3x^2\)[/tex] by each term in the divisor:
- [tex]\(-3x^2 \times x^3 = -3x^5\)[/tex]
- [tex]\(-3x^2 \times 6x^2 = -18x^4\)[/tex]
- [tex]\(-3x^2 \times -3x = 9x^3\)[/tex]
- [tex]\(-3x^2 \times -5 = 15x^2\)[/tex]
Step 4: Subtract the result from the original polynomial.
Perform the subtraction:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This simplifies to:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
Step 5: Repeat the process with the new polynomial.
Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(-4x\)[/tex].
Step 6: Multiply the entire divisor by this new quotient term.
Multiply [tex]\(-4x\)[/tex] by each term in the divisor:
- [tex]\(-4x \times x^3 = -4x^4\)[/tex]
- [tex]\(-4x \times 6x^2 = -24x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times -5 = 20x\)[/tex]
Step 7: Subtract this from the current polynomial.
Perform the subtraction:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
This simplifies to:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
Step 8: Repeat the process with the new polynomial.
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(2\)[/tex].
Step 9: Multiply the entire divisor by this last quotient term.
Multiply [tex]\(2\)[/tex] by each term in the divisor:
- [tex]\(2 \times x^3 = 2x^3\)[/tex]
- [tex]\(2 \times 6x^2 = 12x^2\)[/tex]
- [tex]\(2 \times -3x = -6x\)[/tex]
- [tex]\(2 \times -5 = -10\)[/tex]
Step 10: Subtract to find the remainder.
Perform the subtraction:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
This simplifies to:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
So, the quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex].
The final answer is:
[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]
Step 1: Set up the long division.
The dividend is [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor is [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
Step 2: Divide the first term of the dividend by the first term of the divisor.
Divide [tex]\(-3x^5\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(-3x^2\)[/tex].
Step 3: Multiply the entire divisor by this quotient term.
Multiply [tex]\(-3x^2\)[/tex] by each term in the divisor:
- [tex]\(-3x^2 \times x^3 = -3x^5\)[/tex]
- [tex]\(-3x^2 \times 6x^2 = -18x^4\)[/tex]
- [tex]\(-3x^2 \times -3x = 9x^3\)[/tex]
- [tex]\(-3x^2 \times -5 = 15x^2\)[/tex]
Step 4: Subtract the result from the original polynomial.
Perform the subtraction:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This simplifies to:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
Step 5: Repeat the process with the new polynomial.
Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(-4x\)[/tex].
Step 6: Multiply the entire divisor by this new quotient term.
Multiply [tex]\(-4x\)[/tex] by each term in the divisor:
- [tex]\(-4x \times x^3 = -4x^4\)[/tex]
- [tex]\(-4x \times 6x^2 = -24x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times -5 = 20x\)[/tex]
Step 7: Subtract this from the current polynomial.
Perform the subtraction:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
This simplifies to:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
Step 8: Repeat the process with the new polynomial.
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(2\)[/tex].
Step 9: Multiply the entire divisor by this last quotient term.
Multiply [tex]\(2\)[/tex] by each term in the divisor:
- [tex]\(2 \times x^3 = 2x^3\)[/tex]
- [tex]\(2 \times 6x^2 = 12x^2\)[/tex]
- [tex]\(2 \times -3x = -6x\)[/tex]
- [tex]\(2 \times -5 = -10\)[/tex]
Step 10: Subtract to find the remainder.
Perform the subtraction:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
This simplifies to:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
So, the quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex].
The final answer is:
[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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