Thank you for visiting Divide using long division tex frac 3x 5 22x 4 13x 3 39x 2 14x 6 x 3 6x 2 3x 5 tex Show work. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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 termâ€byâ€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 termâ€byâ€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.
Thank you for reading the article Divide using long division tex frac 3x 5 22x 4 13x 3 39x 2 14x 6 x 3 6x 2 3x 5 tex Show work. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
- You are operating a recreational vessel less than 39 4 feet long on federally controlled waters Which of the following is a legal sound device
- Which step should a food worker complete to prevent cross contact when preparing and serving an allergen free meal A Clean and sanitize all surfaces
- For one month Siera calculated her hometown s average high temperature in degrees Fahrenheit She wants to convert that temperature from degrees Fahrenheit to degrees
Rewritten by : Jeany
