Question

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

Answer 1

Step 1: Convert the polynomial in the synthetic division.

Given polynomial is $f\left(x\right)=x^4-4x^3+2x^2-4x+6$.

Synthetic division method is used to divide a polynomial $f\left(x\right)$ with factor $x-c$, where $c$ is a divisor.

Here, we need to find the value of $f\left(-3\right)$.

Let $c=-3$.

Therefore, polynomial $f\left(x\right)=x^4-4x^3+2x^2-4x+6$ in terms of synthetic division method is expressed by writing the coefficients of $x$ inside division symbol with divisor $-3$.

$\begin{array}{c}1-42-46\\ \end{array}-3$

Step 2: Divide $f\left(x\right)=x^4-4x^3+2x^2-4x+6$ using the synthetic division method.

Steps of synthetic division:

a. Drop down the first coefficient.

$\begin{array}{c}1-42-46\\ \end{array}1-3$

b. Multiply the drop-down of the first column by the divisor $-3$ and drop it in the second column and add the values of that column.

$\begin{array}{c}1-42-46\\ -3\end{array}1-7-3$

c. Multiply the drop-down of the second column by the divisor $-3$ and drop it in the third column and add the values of that column.

$\begin{array}{c}1-42-46\\ -321\end{array}1-723-3$

d. Multiply the drop-down of the third column by the divisor $-3$ and drop it in the fourth column and add the values of that column.

$\begin{array}{c}1-42-46\\ -321-69\end{array}1-723-73-3$

e. Multiply the drop-down of the fourth column by the divisor $-3$ and drop it in the fifth column and add the values of that column.

$\begin{array}{c}1-42-46\\ -321-69219\end{array}1-723-73225-3$

Therefore, the remainder is $225$.

Hence, the value of $f\left(-3\right)$ is $225$.