Answer :
We wish to divide
[tex]$$
-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6
$$[/tex]
by
[tex]$$
x^3 + 6x^2 - 3x - 5.
$$[/tex]
We perform the division step by step.
–––––––––––
Step 1:
Divide the leading term of the numerator, [tex]$-3x^5$[/tex], by the leading term of the denominator, [tex]$x^3$[/tex]. This gives
[tex]$$
\frac{-3x^5}{x^3} = -3x^2.
$$[/tex]
Multiply the entire denominator by [tex]$-3x^2$[/tex]:
[tex]$$
-3x^2(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2.
$$[/tex]
Subtract this product from the original numerator:
[tex]\[
\begin{array}{rcl}
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x -6) & - & (-3x^5 - 18x^4 + 9x^3 + 15x^2) \\
& = & (-3x^5 + 3x^5) + (-22x^4 + 18x^4) + (-13x^3 - 9x^3) \\
& & +\; (39x^2 - 15x^2) + 14x - 6 \\
& = & -4x^4 - 22x^3 + 24x^2 + 14x - 6.
\end{array}
\][/tex]
–––––––––––
Step 2:
Now, divide the new leading term, [tex]$-4x^4$[/tex], by the leading term of the denominator, [tex]$x^3$[/tex]. This gives
[tex]$$
\frac{-4x^4}{x^3} = -4x.
$$[/tex]
Multiply the denominator by [tex]$-4x$[/tex]:
[tex]$$
-4x(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x.
$$[/tex]
Subtract this from the previous result:
[tex]\[
\begin{array}{rcl}
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) & - & (-4x^4 - 24x^3 + 12x^2 + 20x) \\
& = & (-4x^4 + 4x^4) + (-22x^3 + 24x^3) + (24x^2 - 12x^2) \\
& & +\; (14x - 20x) - 6 \\
& = & 2x^3 + 12x^2 - 6x - 6.
\end{array}
\][/tex]
–––––––––––
Step 3:
Finally, divide the new leading term, [tex]$2x^3$[/tex], by [tex]$x^3$[/tex]:
[tex]$$
\frac{2x^3}{x^3} = 2.
$$[/tex]
Multiply the denominator by [tex]$2$[/tex]:
[tex]$$
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10.
$$[/tex]
Subtract this product from the previous result:
[tex]\[
\begin{array}{rcl}
(2x^3 + 12x^2 - 6x - 6) & - & (2x^3 + 12x^2 - 6x - 10) \\
& = & (2x^3 - 2x^3) + (12x^2 - 12x^2) + (-6x + 6x) + (-6 + 10) \\
& = & 4.
\end{array}
\][/tex]
–––––––––––
Conclusion:
The quotient obtained from the division is
[tex]$$
-3x^2 - 4x + 2,
$$[/tex]
and the remainder is [tex]$4$[/tex]. Thus, we can write the division in the form
[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]
This completes the long division process.
[tex]$$
-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6
$$[/tex]
by
[tex]$$
x^3 + 6x^2 - 3x - 5.
$$[/tex]
We perform the division step by step.
–––––––––––
Step 1:
Divide the leading term of the numerator, [tex]$-3x^5$[/tex], by the leading term of the denominator, [tex]$x^3$[/tex]. This gives
[tex]$$
\frac{-3x^5}{x^3} = -3x^2.
$$[/tex]
Multiply the entire denominator by [tex]$-3x^2$[/tex]:
[tex]$$
-3x^2(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2.
$$[/tex]
Subtract this product from the original numerator:
[tex]\[
\begin{array}{rcl}
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x -6) & - & (-3x^5 - 18x^4 + 9x^3 + 15x^2) \\
& = & (-3x^5 + 3x^5) + (-22x^4 + 18x^4) + (-13x^3 - 9x^3) \\
& & +\; (39x^2 - 15x^2) + 14x - 6 \\
& = & -4x^4 - 22x^3 + 24x^2 + 14x - 6.
\end{array}
\][/tex]
–––––––––––
Step 2:
Now, divide the new leading term, [tex]$-4x^4$[/tex], by the leading term of the denominator, [tex]$x^3$[/tex]. This gives
[tex]$$
\frac{-4x^4}{x^3} = -4x.
$$[/tex]
Multiply the denominator by [tex]$-4x$[/tex]:
[tex]$$
-4x(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x.
$$[/tex]
Subtract this from the previous result:
[tex]\[
\begin{array}{rcl}
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) & - & (-4x^4 - 24x^3 + 12x^2 + 20x) \\
& = & (-4x^4 + 4x^4) + (-22x^3 + 24x^3) + (24x^2 - 12x^2) \\
& & +\; (14x - 20x) - 6 \\
& = & 2x^3 + 12x^2 - 6x - 6.
\end{array}
\][/tex]
–––––––––––
Step 3:
Finally, divide the new leading term, [tex]$2x^3$[/tex], by [tex]$x^3$[/tex]:
[tex]$$
\frac{2x^3}{x^3} = 2.
$$[/tex]
Multiply the denominator by [tex]$2$[/tex]:
[tex]$$
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10.
$$[/tex]
Subtract this product from the previous result:
[tex]\[
\begin{array}{rcl}
(2x^3 + 12x^2 - 6x - 6) & - & (2x^3 + 12x^2 - 6x - 10) \\
& = & (2x^3 - 2x^3) + (12x^2 - 12x^2) + (-6x + 6x) + (-6 + 10) \\
& = & 4.
\end{array}
\][/tex]
–––––––––––
Conclusion:
The quotient obtained from the division is
[tex]$$
-3x^2 - 4x + 2,
$$[/tex]
and the remainder is [tex]$4$[/tex]. Thus, we can write the division in the form
[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]
This completes the long division process.
