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Answer :
Sure! Let's perform polynomial long division to divide [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
### Step-by-Step Solution:
1. Setup the Division:
Write the dividend [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] under the division bar 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]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This is the first term of the quotient.
3. Multiply and Subtract:
Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract the result from the original dividend:
[tex]\[
(-3x^2) \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This simplifies to:
[tex]\[
-4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the Process:
- Divide the new leading term, [tex]\(-4x^4\)[/tex], by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Add [tex]\(-4x\)[/tex] to the quotient.
- Multiply the entire divisor by [tex]\(-4x\)[/tex] and subtract:
[tex]\[
(-4x) \cdot (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
This simplifies to:
[tex]\[
2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue Until Degree is Lower:
- Divide the leading term [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Add [tex]\(2\)[/tex] to the quotient.
- Multiply the entire divisor by [tex]\(2\)[/tex] and subtract:
[tex]\[
2 \cdot (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
This simplifies to:
[tex]\[
4
\][/tex]
6. Result of Division:
- The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex].
- The remainder is [tex]\(4\)[/tex].
### Final Answer:
The result of the long division is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
And that's how you perform polynomial long division step-by-step!
### Step-by-Step Solution:
1. Setup the Division:
Write the dividend [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] under the division bar 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]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This is the first term of the quotient.
3. Multiply and Subtract:
Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract the result from the original dividend:
[tex]\[
(-3x^2) \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This simplifies to:
[tex]\[
-4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the Process:
- Divide the new leading term, [tex]\(-4x^4\)[/tex], by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Add [tex]\(-4x\)[/tex] to the quotient.
- Multiply the entire divisor by [tex]\(-4x\)[/tex] and subtract:
[tex]\[
(-4x) \cdot (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
This simplifies to:
[tex]\[
2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue Until Degree is Lower:
- Divide the leading term [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Add [tex]\(2\)[/tex] to the quotient.
- Multiply the entire divisor by [tex]\(2\)[/tex] and subtract:
[tex]\[
2 \cdot (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
This simplifies to:
[tex]\[
4
\][/tex]
6. Result of Division:
- The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex].
- The remainder is [tex]\(4\)[/tex].
### Final Answer:
The result of the long division is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
And that's how you perform polynomial long division step-by-step!
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