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

Jul 30, 2015

$\textcolor{red}{f \left(- 2\right) = 70}$

The Remainder Theorem states that when we divide a polynomial $f \left(x\right)$ by $x - c$ the remainder $R$ equals $f \left(c\right)$.

We use synthetic division to divide $f \left(x\right)$ by $x - c$, where $c = - 2$.

Step 1. Write only the coefficients of $x$ in the dividend inside an upside-down division symbol.

$| 1 \text{ " -4" " "2" " " " "-4" " " " } 6$
$| \textcolor{w h i t e}{1}$
stackrel("————————————————————)

Step 2. Put the divisor at the left.

$\textcolor{red}{- 2} | 1 \text{ " -4" " "2" " " " "-4" " " " } 6$
$\text{ " } |$
" "stackrel("————————————————————)

Step 3. Drop the first coefficient of the dividend below the division symbol.

$- 2 | 1 \text{ " -4" " "2" " " " "-4" " " " } 6$
$\text{ " } |$
" "stackrel("————————————————————)
$\text{ " " } \textcolor{red}{1}$

Step 4. Multiply the drop-down by the divisor, and put the result in the next column.

$- 2 | 1 \text{ " -4" " " " "2"" "-4" } 6$
$\text{ " " |" " " " } \textcolor{red}{- 2}$
" " " "stackrel("————————————————————")
$\text{ " " } 1$

Step 5. Add down the column.

$- 3 | 1 \text{ " -4" " " " "2"" "-4" } 6$
$\text{ " " |" " " " } - 2$
" " " "stackrel("————————————————————)
$\text{ " " "1" " " } \textcolor{red}{- 6}$

Step 6. Repeat Steps 4 and 5 until you can go no farther

$- 2 | 1 \text{ " -4" " "2" " " " "-4" " " " } 6$
$\text{ " " |" " " " "-2" "12" "-28" " " " } 64$
" " " "stackrel("————————————————————)
$\text{ " " "1" "-6" " 14" "-32 " " " } \textcolor{red}{70}$

The remainder is $70$, so $f \left(- 2\right) = 70$.

Check:

x^4-4x^3 +2x^2 –4x +6 = (-2)^4-4(-2)^3+2(-2)^2-4(-2)+6 = 16+32 +8+8+6 = 70