Answer :
Sure! Let's divide the polynomial [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] using long division.
### Step-by-Step Solution:
1. Write the division setup:
[tex]\[
\begin{array}{r|rrrrr}
& -3x^2 & -4x & +2 \\
\hline
x^3 + 6x^2 - 3x - 5 & -3x^5 & -22x^4 & -13x^3 & +39x^2 & +14x & -6 \\
\end{array}
\][/tex]
2. Divide the first terms: [tex]\(-3x^5\)[/tex] (from the dividend) by [tex]\(x^3\)[/tex] (from the divisor), which gives [tex]\(-3x^2\)[/tex].
3. Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
\begin{array}{r}
-3x^2(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\end{array}
\][/tex]
4. Subtract this from the original dividend:
[tex]\[
\begin{array}{rrrrr}
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) \\
- (-3x^5 - 18x^4 + 9x^3 + 15x^2) \\
\hline
0 & -4x^4 & -22x^3 & +24x^2 & +14x & -6
\end{array}
\][/tex]
5. Divide the new leading term: [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(-4x\)[/tex].
6. Multiply the entire divisor by [tex]\(-4x\)[/tex]:
[tex]\[
\begin{array}{r}
-4x(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\end{array}
\][/tex]
7. Subtract this from the current dividend:
[tex]\[
\begin{array}{rrrrr}
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) \\
- (-4x^4 - 24x^3 + 12x^2 + 20x) \\
\hline
0 & 2x^3 & 12x^2 & -6x & -6
\end{array}
\][/tex]
8. Divide the next leading term: [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(2\)[/tex].
9. Multiply the entire divisor by [tex]\(2\)[/tex]:
[tex]\[
\begin{array}{r}
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\end{array}
\][/tex]
10. Subtract this from the current dividend:
[tex]\[
\begin{array}{rrrr}
(2x^3 + 12x^2 - 6x - 6) \\
- (2x^3 + 12x^2 - 6x - 10) \\
\hline
0 & 0 & 0 & +4
\end{array}
\][/tex]
### Final Result:
After performing these steps, we get the quotient to be:
[tex]\[
-3x^2 - 4x + 2
\][/tex]
And the remainder is:
[tex]\[
4
\][/tex]
So the division of [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
### Step-by-Step Solution:
1. Write the division setup:
[tex]\[
\begin{array}{r|rrrrr}
& -3x^2 & -4x & +2 \\
\hline
x^3 + 6x^2 - 3x - 5 & -3x^5 & -22x^4 & -13x^3 & +39x^2 & +14x & -6 \\
\end{array}
\][/tex]
2. Divide the first terms: [tex]\(-3x^5\)[/tex] (from the dividend) by [tex]\(x^3\)[/tex] (from the divisor), which gives [tex]\(-3x^2\)[/tex].
3. Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
\begin{array}{r}
-3x^2(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\end{array}
\][/tex]
4. Subtract this from the original dividend:
[tex]\[
\begin{array}{rrrrr}
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) \\
- (-3x^5 - 18x^4 + 9x^3 + 15x^2) \\
\hline
0 & -4x^4 & -22x^3 & +24x^2 & +14x & -6
\end{array}
\][/tex]
5. Divide the new leading term: [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(-4x\)[/tex].
6. Multiply the entire divisor by [tex]\(-4x\)[/tex]:
[tex]\[
\begin{array}{r}
-4x(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\end{array}
\][/tex]
7. Subtract this from the current dividend:
[tex]\[
\begin{array}{rrrrr}
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) \\
- (-4x^4 - 24x^3 + 12x^2 + 20x) \\
\hline
0 & 2x^3 & 12x^2 & -6x & -6
\end{array}
\][/tex]
8. Divide the next leading term: [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(2\)[/tex].
9. Multiply the entire divisor by [tex]\(2\)[/tex]:
[tex]\[
\begin{array}{r}
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\end{array}
\][/tex]
10. Subtract this from the current dividend:
[tex]\[
\begin{array}{rrrr}
(2x^3 + 12x^2 - 6x - 6) \\
- (2x^3 + 12x^2 - 6x - 10) \\
\hline
0 & 0 & 0 & +4
\end{array}
\][/tex]
### Final Result:
After performing these steps, we get the quotient to be:
[tex]\[
-3x^2 - 4x + 2
\][/tex]
And the remainder is:
[tex]\[
4
\][/tex]
So the division of [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
