Answer :
Sure! Let's perform the long division of 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].
### Step-by-Step Long Division
1. Set Up the Division:
- Dividend: [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex]
- Divisor: [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex]
2. Divide the First Terms:
- Divide the leading term of the dividend by the leading term of the divisor:
[tex]\(\frac{-3x^5}{x^3} = -3x^2\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
-3x^2 \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
- Subtract this from the dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
Resulting in:
[tex]\[
0x^5 + (-4x^4) - 22x^3 + 24x^2 + 14x - 6
\][/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].
5. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(-4x\)[/tex]:
[tex]\[
-4x \cdot (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
- Subtract this from the current remainder:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
Resulting in:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
6. Divide Again:
- Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\(\frac{2x^3}{x^3} = 2\)[/tex].
7. Multiply and Final Subtract:
- Multiply the entire divisor by [tex]\(2\)[/tex]:
[tex]\[
2 \cdot (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
- Subtract this from the current remainder:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
Resulting in:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
### Final Result
- Quotient: [tex]\(-3x^2 - 4x + 2\)[/tex]
- Remainder: [tex]\(4\)[/tex]
Therefore, the quotient of the division is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex], meaning:
[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-by-Step Long Division
1. Set Up the Division:
- Dividend: [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex]
- Divisor: [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex]
2. Divide the First Terms:
- Divide the leading term of the dividend by the leading term of the divisor:
[tex]\(\frac{-3x^5}{x^3} = -3x^2\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
-3x^2 \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
- Subtract this from the dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
Resulting in:
[tex]\[
0x^5 + (-4x^4) - 22x^3 + 24x^2 + 14x - 6
\][/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].
5. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(-4x\)[/tex]:
[tex]\[
-4x \cdot (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
- Subtract this from the current remainder:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
Resulting in:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
6. Divide Again:
- Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\(\frac{2x^3}{x^3} = 2\)[/tex].
7. Multiply and Final Subtract:
- Multiply the entire divisor by [tex]\(2\)[/tex]:
[tex]\[
2 \cdot (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
- Subtract this from the current remainder:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
Resulting in:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
### Final Result
- Quotient: [tex]\(-3x^2 - 4x + 2\)[/tex]
- Remainder: [tex]\(4\)[/tex]
Therefore, the quotient of the division is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex], meaning:
[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]
