# How do you use synthetic substitution to evaluate f(-2) for f(x)=x^4-4x^3-4x+6?

Dec 16, 2015

(see below for method of evaluation)
$f \left(- 2\right) = 62$

#### Explanation:

Evaluation of $f \left(x\right) = \textcolor{b l u e}{1} {x}^{4} \textcolor{b l u e}{- 4} {x}^{3} \textcolor{b l u e}{+ 0} {x}^{2} \textcolor{b l u e}{- 4} x \textcolor{b l u e}{+ 6}$
for $f \left(\textcolor{red}{- 2}\right)$
Notice the expansion of the expression to include the implied coefficients of $\textcolor{b l u e}{1}$ for ${x}^{4}$ and of $\textcolor{b l u e}{0}$ for ${x}^{2}$

Initial set-up:
{: (,"|",color(blue)(1),color(blue)(-4),color(blue)(+0),color(blue)(-4),color(blue)(+6),color(white)("XXXXXXXX")"line "), (,"|",,,,,,color(white)("XXXXXXXX")"line"), ("----",,"----","----","----","----","----",), (xxcolor(red)((-2)),"|",color(green)(1),,,,,color(white)("XXXXXXXX")"line ") :}

For each column

• Multiply the last number written on line  by $\textcolor{red}{\left(- 2\right)}$ and write the product on line  of the next column
• Add the numbers in lines  and  of the next column and write the sum in line  of that column.

The number written in the last column of line  will be the value of $f \left(\textcolor{red}{- 2}\right)$

{: (,"|",color(blue)(1),color(blue)(-4),color(blue)(+0),color(blue)(-4),color(blue)(+6),color(white)("XXXXXXXX")"line "), (,"|",,-2,12,-24,56,color(white)("XXXXXXXX")"line"), ("-----------",,"----","----","----","----","----",), (xxcolor(red)((-2)),"|",color(green)(1),-6,12,-28,color(cyan)(62),color(white)("XXXXXXXX")"line ") :}