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

##### 1 Answer
Jul 30, 2015

$\textcolor{red}{f \left(- 3\right) = 225}$

#### Explanation:

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 $c = - 3$.

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.

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

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

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

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

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

Step 5. Add down the column.

$- 3 | 1 \text{ " -4" " " " "2"" "-4" } 6$
$\text{ " " |" " " " } - 3$
" "stackrel("————————————————————)
$\text{ " " "1" " " } \textcolor{red}{- 7}$

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

$- 3 | 1 \text{ " -4" " " " "2"" "-4" " " " " } 6$
$\text{ " " |" " " " "-3" "+21 -69" " " } 219$
" "stackrel("————————————————————)
$\text{ " " "1" " -7" " " " 23" "-73 " " } \textcolor{red}{225}$

The remainder is $225$, so $f \left(- 3\right) = 225$.

Check:

x^4-4x^3 +2x^2 –4x +6 = (-3)^4-4(-3)^3+2(-3)^2-4(-3)+6 = 81+108 +18+12+6 = 225