# How do you use synthetic division to divide (x^2+4x+1) by x-5?

Feb 11, 2016

$x + 9 + \frac{46}{x - 5}$

For explanation, see below.

#### Explanation:

Dividing by $x - 5$ means that the corresponding root of $5$ will be in the upper left of the synthetic division. The numbers up top will be the coefficients of ${x}^{2} + 4 x + 1$, which is $1 , 4 , 1$.

$\left.\begin{matrix}\underline{5} \text{|" & " 1" & " 4" & " 1" & " " \\ ul" " & ul" " & ul" " & ul" " & + \\ " " & " " & " " & "| " & " }\end{matrix}\right.$

Move the $1$ in the first column down to the bottom.

$\left.\begin{matrix}\underline{5} \text{|" & " 1" & " 4" & " 1" & " " \\ ul" " & ul" " & ul" " & ul" " & + \\ " " & " 1" & " " & "| " & " }\end{matrix}\right.$

Multiply the $1$ by the $5$ in the corner for a product of $5$, which goes in the next column.

$\left.\begin{matrix}\underline{5} \text{|" & " 1" & " "4 & " 1" & " " \\ ul" " & ul" " & ul(" "color(red)5) & ul" " & + \\ " " & " "color(red)1 & " " & "| " & " }\end{matrix}\right.$

Add the $4$ and the $5$ in the new column.

$\left.\begin{matrix}\underline{5} \text{|" & " 1" & " "4 & " 1" & " " \\ ul" " & ul" " & ul(" "color(red)5) & ul" " & + \\ " " & " "color(red)1 & " "9 & "| " & " }\end{matrix}\right.$

Multiply the $9$ by the $5$ in the corner for $45$, which goes in the next column.

$\left.\begin{matrix}\underline{5} \text{|" & " 1" & " "4 & " 1" & " " \\ ul" " & ul" " & ul(" "color(red)5) & ul(" "color(blue)(45)) & + \\ " " & " "color(red)1 & " "color(blue)9 & "| " & " }\end{matrix}\right.$

Add the $1$ and $45$. This goes in the "remainder" spot.

$\left.\begin{matrix}\underline{5} \text{|" & " 1" & " "4 & " 1" & " " \\ ul" " & ul" " & ul(" "color(red)5) & ul(" "color(blue)(45)) & + \\ " " & " "color(red)1 & " "color(blue)9 & "|46" & " }\end{matrix}\right.$

The numbers $1 , 9$ in the bottom row correlate to the coefficients of the quotient. The $46$ is the numerator of the remainder quotient:

$\frac{{x}^{2} + 4 x + 1}{x - 5} = x + 9 + \frac{46}{x - 5}$