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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]
The goal is to express the division as
[tex]$$
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = Q(x) + \frac{R(x)}{x^3 + 6x^2 - 3x - 5},
$$[/tex]
where [tex]$Q(x)$[/tex] is the quotient and [tex]$R(x)$[/tex] is the remainder with a degree less than [tex]$3$[/tex].
Let’s perform the long division step by step.
–––––– Step 1: Divide the leading terms
The leading term of the dividend is [tex]$-3x^5$[/tex] and that of the divisor is [tex]$x^3$[/tex]. Dividing these gives:
[tex]$$
\frac{-3x^5}{x^3} = -3x^2.
$$[/tex]
So, the first term of the quotient is [tex]$-3x^2$[/tex].
–––––– Step 2: Multiply and subtract
Multiply the entire divisor 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 from the original dividend:
[tex]\[
\begin{array}{rrrrrr}
& -3x^5 & -22x^4 & -13x^3 & +39x^2 & +14x -6\\[5mm]
-&( -3x^5 & -18x^4 & +9x^3 & +15x^2 & \quad )\\ \hline
& \quad 0 & -4x^4 & -22x^3 & +24x^2 & +14x -6\\
\end{array}
\][/tex]
After subtraction, the new polynomial is:
[tex]$$
-4x^4 - 22x^3 + 24x^2 + 14x - 6.
$$[/tex]
–––––– Step 3: Repeat with the new polynomial
Now, divide the leading term of the new polynomial ([tex]$-4x^4$[/tex]) by the leading term of the divisor ([tex]$x^3$[/tex]):
[tex]$$
\frac{-4x^4}{x^3} = -4x.
$$[/tex]
This gives the next term of the quotient: [tex]$-4x$[/tex].
Multiply the divisor by [tex]$-4x$[/tex]:
[tex]$$
-4x(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x.
$$[/tex]
Subtract this from the current polynomial:
[tex]\[
\begin{array}{rrrrrr}
& -4x^4 & -22x^3 & +24x^2 & +14x & -6 \\[5mm]
-&( -4x^4 & -24x^3 & +12x^2 & +20x & \quad )\\ \hline
& \quad 0 & \;2x^3 & +12x^2 & -6x & -6\\
\end{array}
\][/tex]
The result is:
[tex]$$
2x^3 + 12x^2 - 6x - 6.
$$[/tex]
–––––– Step 4: One more division step
Now, divide the leading term of the new polynomial ([tex]$2x^3$[/tex]) by the leading term of the divisor ([tex]$x^3$[/tex]):
[tex]$$
\frac{2x^3}{x^3} = 2.
$$[/tex]
So, the next term of the quotient is [tex]$2$[/tex].
Multiply the divisor by [tex]$2$[/tex]:
[tex]$$
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10.
$$[/tex]
Subtract to find the remainder:
[tex]\[
\begin{array}{rrrrrr}
& 2x^3 & +12x^2 & -6x & -6 \\[5mm]
-&( 2x^3 & +12x^2 & -6x & -10)\\ \hline
& \quad 0 & \quad 0 & \;0 & \;4\\
\end{array}
\][/tex]
Thus, the remainder is [tex]$4$[/tex].
–––––– Final Result
The quotient is
[tex]$$
Q(x) = -3x^2 - 4x + 2,
$$[/tex]
and the remainder is
[tex]$$
R(x) = 4.
$$[/tex]
We can now express the original division as:
[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 step by step.
[tex]$$
-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6
$$[/tex]
by
[tex]$$
x^3 + 6x^2 - 3x - 5.
$$[/tex]
The goal is to express the division as
[tex]$$
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = Q(x) + \frac{R(x)}{x^3 + 6x^2 - 3x - 5},
$$[/tex]
where [tex]$Q(x)$[/tex] is the quotient and [tex]$R(x)$[/tex] is the remainder with a degree less than [tex]$3$[/tex].
Let’s perform the long division step by step.
–––––– Step 1: Divide the leading terms
The leading term of the dividend is [tex]$-3x^5$[/tex] and that of the divisor is [tex]$x^3$[/tex]. Dividing these gives:
[tex]$$
\frac{-3x^5}{x^3} = -3x^2.
$$[/tex]
So, the first term of the quotient is [tex]$-3x^2$[/tex].
–––––– Step 2: Multiply and subtract
Multiply the entire divisor 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 from the original dividend:
[tex]\[
\begin{array}{rrrrrr}
& -3x^5 & -22x^4 & -13x^3 & +39x^2 & +14x -6\\[5mm]
-&( -3x^5 & -18x^4 & +9x^3 & +15x^2 & \quad )\\ \hline
& \quad 0 & -4x^4 & -22x^3 & +24x^2 & +14x -6\\
\end{array}
\][/tex]
After subtraction, the new polynomial is:
[tex]$$
-4x^4 - 22x^3 + 24x^2 + 14x - 6.
$$[/tex]
–––––– Step 3: Repeat with the new polynomial
Now, divide the leading term of the new polynomial ([tex]$-4x^4$[/tex]) by the leading term of the divisor ([tex]$x^3$[/tex]):
[tex]$$
\frac{-4x^4}{x^3} = -4x.
$$[/tex]
This gives the next term of the quotient: [tex]$-4x$[/tex].
Multiply the divisor by [tex]$-4x$[/tex]:
[tex]$$
-4x(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x.
$$[/tex]
Subtract this from the current polynomial:
[tex]\[
\begin{array}{rrrrrr}
& -4x^4 & -22x^3 & +24x^2 & +14x & -6 \\[5mm]
-&( -4x^4 & -24x^3 & +12x^2 & +20x & \quad )\\ \hline
& \quad 0 & \;2x^3 & +12x^2 & -6x & -6\\
\end{array}
\][/tex]
The result is:
[tex]$$
2x^3 + 12x^2 - 6x - 6.
$$[/tex]
–––––– Step 4: One more division step
Now, divide the leading term of the new polynomial ([tex]$2x^3$[/tex]) by the leading term of the divisor ([tex]$x^3$[/tex]):
[tex]$$
\frac{2x^3}{x^3} = 2.
$$[/tex]
So, the next term of the quotient is [tex]$2$[/tex].
Multiply the divisor by [tex]$2$[/tex]:
[tex]$$
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10.
$$[/tex]
Subtract to find the remainder:
[tex]\[
\begin{array}{rrrrrr}
& 2x^3 & +12x^2 & -6x & -6 \\[5mm]
-&( 2x^3 & +12x^2 & -6x & -10)\\ \hline
& \quad 0 & \quad 0 & \;0 & \;4\\
\end{array}
\][/tex]
Thus, the remainder is [tex]$4$[/tex].
–––––– Final Result
The quotient is
[tex]$$
Q(x) = -3x^2 - 4x + 2,
$$[/tex]
and the remainder is
[tex]$$
R(x) = 4.
$$[/tex]
We can now express the original division as:
[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 step by step.
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