# 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 [1]"), (,"|",,,,,,color(white)("XXXXXXXX")"line[2]"), ("----",,"----","----","----","----","----",), (xxcolor(red)((-2)),"|",color(green)(1),,,,,color(white)("XXXXXXXX")"line [3]") :}

For each column

• Multiply the last number written on line [3] by $\textcolor{red}{\left(- 2\right)}$ and write the product on line [2] of the next column
• Add the numbers in lines [1] and [2] of the next column and write the sum in line [3] of that column.

The number written in the last column of line [3] 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 [1]"), (,"|",,-2,12,-24,56,color(white)("XXXXXXXX")"line[2]"), ("-----------",,"----","----","----","----","----",), (xxcolor(red)((-2)),"|",color(green)(1),-6,12,-28,color(cyan)(62),color(white)("XXXXXXXX")"line [3]") :}