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 will do so by long division.
–––––––––––––––––––––––
Step 1. Determine the first term of the quotient
Divide the leading term of the numerator by the leading term of the divisor:
[tex]$$
\frac{3x^5}{x^3} = 3x^2.
$$[/tex]
Multiply the entire divisor by [tex]$3x^2$[/tex]:
[tex]$$
3x^2 \cdot \left(x^3 + 6x^2 - 3x + 5\right) = 3x^5 + 18x^4 - 9x^3 + 15x^2.
$$[/tex]
Subtract this product from the original numerator:
[tex]\[
\begin{array}{rcl}
\text{Numerator} &=& 3x^5 - 22x^4 + 13x^3 + 39x^2 + 14x + 6,\\[1mm]
\text{Product} &=& 3x^5 + 18x^4 - 9x^3 + 15x^2.
\end{array}
\][/tex]
Performing the subtraction:
[tex]\[
\begin{aligned}
3x^5 - 3x^5 &= 0,\\[1mm]
-22x^4 - 18x^4 &= -40x^4,\\[1mm]
13x^3 - (-9x^3) &= 22x^3,\\[1mm]
39x^2 - 15x^2 &= 24x^2,\\[1mm]
\text{Bring down } 14x + 6.
\end{aligned}
\][/tex]
So, the new remainder is
[tex]$$
-40x^4 + 22x^3 + 24x^2 + 14x + 6.
$$[/tex]
–––––––––––––––––––––––
Step 2. Find the next term of the quotient
Divide the new remainder’s leading term by the divisor’s leading term:
[tex]$$
\frac{-40x^4}{x^3} = -40x.
$$[/tex]
Multiply the divisor by [tex]$-40x$[/tex]:
[tex]$$
-40x \cdot \left(x^3 + 6x^2 - 3x + 5\right) = -40x^4 - 240x^3 + 120x^2 - 200x.
$$[/tex]
Subtract this from the previous remainder:
[tex]\[
\begin{array}{rcl}
\text{Previous remainder} &=& -40x^4 + 22x^3 + 24x^2 + 14x + 6,\\[1mm]
\text{Product} &=& -40x^4 - 240x^3 + 120x^2 - 200x.
\end{array}
\][/tex]
Subtracting gives:
[tex]\[
\begin{aligned}
-40x^4 - (-40x^4) &= 0,\\[1mm]
22x^3 - (-240x^3) &= 262x^3,\\[1mm]
24x^2 - 120x^2 &= -96x^2,\\[1mm]
14x - (-200x) &= 214x,\\[1mm]
\text{Bring down } + 6.
\end{aligned}
\][/tex]
Now the remainder becomes
[tex]$$
262x^3 - 96x^2 + 214x + 6.
$$[/tex]
–––––––––––––––––––––––
Step 3. Find the next term of the quotient
Divide the leading term of the current remainder by [tex]$x^3$[/tex]:
[tex]$$
\frac{262x^3}{x^3} = 262.
$$[/tex]
Multiply the divisor by [tex]$262$[/tex]:
[tex]$$
262 \cdot \left(x^3 + 6x^2 - 3x + 5\right) = 262x^3 + 1572x^2 - 786x + 1310.
$$[/tex]
Subtract this product from the current remainder:
[tex]\[
\begin{array}{rcl}
\text{Current remainder} &=& 262x^3 - 96x^2 + 214x + 6,\\[1mm]
\text{Product} &=& 262x^3 + 1572x^2 - 786x + 1310.
\end{array}
\][/tex]
Subtracting term‐by‐term:
[tex]\[
\begin{aligned}
262x^3 - 262x^3 &= 0,\\[1mm]
-96x^2 - 1572x^2 &= -1668x^2,\\[1mm]
214x - (-786x) &= 1000x,\\[1mm]
6 - 1310 &= -1304.
\end{aligned}
\][/tex]
So, the final remainder is
[tex]$$
-1668x^2 + 1000x - 1304.
$$[/tex]
–––––––––––––––––––––––
Final Result
The quotient obtained is the sum of the individual quotient terms:
[tex]$$
\text{Quotient} = 3x^2 - 40x + 262,
$$[/tex]
and the remainder is
[tex]$$
-1668x^2 + 1000x - 1304.
$$[/tex]
Thus, the division can be written as:
[tex]$$
\frac{3x^5 - 22x^4 + 13x^3 + 39x^2 + 14x + 6}{x^3 + 6x^2 - 3x + 5} = 3x^2 - 40x + 262 + \frac{-1668x^2 + 1000x - 1304}{x^3 + 6x^2 - 3x + 5}.
$$[/tex]
This is the complete step-by-step solution of the long division.
[tex]$$
3x^5 - 22x^4 + 13x^3 + 39x^2 + 14x + 6
$$[/tex]
by
[tex]$$
x^3 + 6x^2 - 3x + 5.
$$[/tex]
We will do so by long division.
–––––––––––––––––––––––
Step 1. Determine the first term of the quotient
Divide the leading term of the numerator by the leading term of the divisor:
[tex]$$
\frac{3x^5}{x^3} = 3x^2.
$$[/tex]
Multiply the entire divisor by [tex]$3x^2$[/tex]:
[tex]$$
3x^2 \cdot \left(x^3 + 6x^2 - 3x + 5\right) = 3x^5 + 18x^4 - 9x^3 + 15x^2.
$$[/tex]
Subtract this product from the original numerator:
[tex]\[
\begin{array}{rcl}
\text{Numerator} &=& 3x^5 - 22x^4 + 13x^3 + 39x^2 + 14x + 6,\\[1mm]
\text{Product} &=& 3x^5 + 18x^4 - 9x^3 + 15x^2.
\end{array}
\][/tex]
Performing the subtraction:
[tex]\[
\begin{aligned}
3x^5 - 3x^5 &= 0,\\[1mm]
-22x^4 - 18x^4 &= -40x^4,\\[1mm]
13x^3 - (-9x^3) &= 22x^3,\\[1mm]
39x^2 - 15x^2 &= 24x^2,\\[1mm]
\text{Bring down } 14x + 6.
\end{aligned}
\][/tex]
So, the new remainder is
[tex]$$
-40x^4 + 22x^3 + 24x^2 + 14x + 6.
$$[/tex]
–––––––––––––––––––––––
Step 2. Find the next term of the quotient
Divide the new remainder’s leading term by the divisor’s leading term:
[tex]$$
\frac{-40x^4}{x^3} = -40x.
$$[/tex]
Multiply the divisor by [tex]$-40x$[/tex]:
[tex]$$
-40x \cdot \left(x^3 + 6x^2 - 3x + 5\right) = -40x^4 - 240x^3 + 120x^2 - 200x.
$$[/tex]
Subtract this from the previous remainder:
[tex]\[
\begin{array}{rcl}
\text{Previous remainder} &=& -40x^4 + 22x^3 + 24x^2 + 14x + 6,\\[1mm]
\text{Product} &=& -40x^4 - 240x^3 + 120x^2 - 200x.
\end{array}
\][/tex]
Subtracting gives:
[tex]\[
\begin{aligned}
-40x^4 - (-40x^4) &= 0,\\[1mm]
22x^3 - (-240x^3) &= 262x^3,\\[1mm]
24x^2 - 120x^2 &= -96x^2,\\[1mm]
14x - (-200x) &= 214x,\\[1mm]
\text{Bring down } + 6.
\end{aligned}
\][/tex]
Now the remainder becomes
[tex]$$
262x^3 - 96x^2 + 214x + 6.
$$[/tex]
–––––––––––––––––––––––
Step 3. Find the next term of the quotient
Divide the leading term of the current remainder by [tex]$x^3$[/tex]:
[tex]$$
\frac{262x^3}{x^3} = 262.
$$[/tex]
Multiply the divisor by [tex]$262$[/tex]:
[tex]$$
262 \cdot \left(x^3 + 6x^2 - 3x + 5\right) = 262x^3 + 1572x^2 - 786x + 1310.
$$[/tex]
Subtract this product from the current remainder:
[tex]\[
\begin{array}{rcl}
\text{Current remainder} &=& 262x^3 - 96x^2 + 214x + 6,\\[1mm]
\text{Product} &=& 262x^3 + 1572x^2 - 786x + 1310.
\end{array}
\][/tex]
Subtracting term‐by‐term:
[tex]\[
\begin{aligned}
262x^3 - 262x^3 &= 0,\\[1mm]
-96x^2 - 1572x^2 &= -1668x^2,\\[1mm]
214x - (-786x) &= 1000x,\\[1mm]
6 - 1310 &= -1304.
\end{aligned}
\][/tex]
So, the final remainder is
[tex]$$
-1668x^2 + 1000x - 1304.
$$[/tex]
–––––––––––––––––––––––
Final Result
The quotient obtained is the sum of the individual quotient terms:
[tex]$$
\text{Quotient} = 3x^2 - 40x + 262,
$$[/tex]
and the remainder is
[tex]$$
-1668x^2 + 1000x - 1304.
$$[/tex]
Thus, the division can be written as:
[tex]$$
\frac{3x^5 - 22x^4 + 13x^3 + 39x^2 + 14x + 6}{x^3 + 6x^2 - 3x + 5} = 3x^2 - 40x + 262 + \frac{-1668x^2 + 1000x - 1304}{x^3 + 6x^2 - 3x + 5}.
$$[/tex]
This is the complete step-by-step solution of the long division.
