# How do you use synthetic division to divide (x^7 - x^6 + x^5 - x^4 + 2) / (x + 1)?

Nov 20, 2015

${x}^{6} - 2 {x}^{5} + 3 {x}^{4} - 4 {x}^{3} + 4 {x}^{2} - 4 {x}^{1} + 4$ with remainder $= \left(- 2\right)$

#### Explanation:

Write the dividend expression in order of descending powers of $x$ including all powers of $x$ showing all coefficient values even those with coefficients of $0$

Given dividend expression becomes:
$\textcolor{w h i t e}{\text{XXX}} 1 {x}^{7} + \left(- 1\right) {x}^{6} + 1 {x}^{5} + \left(- 1\right) {x}^{4} + 0 {x}^{3} + 0 {x}^{2} + {x}^{1} + \left(- 2\right) {x}^{0}$

Setup for synthetic division (by monic binomial)
Write the coefficients of the dividend expression as a row (line 1).
Leave a blank line or a line with just a reminder $+$ sign (line 2)
Draw a separator line (optional)
Copy the$\textcolor{b l u e}{\text{ value of the first coefficient}}$ in the column under the first coefficient to the bottom line; you may, as I have done put a prefix on this line indicating a multiplication by $\textcolor{g r e e n}{\text{the negative of the constant term of the divisor}}$.
You should have something like below (note that elements in $\textcolor{b r o w n}{\text{brown}}$ are for reference purposes only; elements in $\textcolor{g r e e n}{\text{green}}$ are optional - recommended).

{: (,,,color(brown)(x^7),color(brown)(x^6),color(brown)(x^5),color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0)), (color(brown)("line 1"),,"|",1,-1,+1,-1,0,0,0,-2), (color(brown)("line 2"),+,"|",,,,,,,,), (,,,"-----","-----","-----","-----","-----","-----","-----","-----"), (color(brown)("line 3"),xxcolor(green)((-1)),"|",color(blue)(1),,,,,,,) :}

Process Steps:
1. Multiply the value in the last completed column by $\textcolor{g r e e n}{\text{the negative of the constant term of the divisor}}$ and write the product on color(brown)("line 2") in the next column.
2. Move to the next column and add the coefficient (from $\textcolor{b r o w n}{\text{line 1}}$ and the product produced in step 1; write the sum in this column of $\textcolor{b r o w n}{\text{line 3}}$.
Repeat these steps until you have written a sum in the right-most column.

You should have something like:

{: (,,,color(brown)(x^7),color(brown)(x^6),color(brown)(x^5),color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0)), (color(brown)("line 1"),,"|",1,-1,+1,-1,0,0,0,-2), (color(brown)("line 2"),+,"|",,-1,+2,-3,+4,-4,+4,-4), (,,,"-----","-----","-----","-----","-----","-----","-----","-----"), (color(brown)("line 3"),xxcolor(green)((-1)),"|",color(blue)(1),color(blue)(-2),color(blue)(+3),color(blue)(-4),color(blue)(+4),color(blue)(-4),color(blue)(+4),color(red)(-2)), (,,,color(brown)(x^6),color(brown)(x^5),color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0),color(brown)("R")) :}

The $\textcolor{red}{\text{final sum}}$ is the Remainder;
the $\textcolor{b l u e}{\text{preceding sums}}$ are the coefficients of the reduced quotient polynomial.