Question

Divide using synthetic division. Write down the answer as a polynomial.x^3-5x^2-2x+24=0; (x+2)

Answer 1

we are given the following polynomial:

$x^3-5x^2-2x+24=0$

we are asked to use synthetic division by:

$x+2$

first we need to find the root of "x + 2":

$\begin{gathered} x+2=0 \\ x=-2 \end{gathered}$

Now we do the synthetic division using the following array:

$\begin{bmatrix}{1} & {-5} & {-2} \\ {\square} & {\square} & {\square} \\ {\square} & {\square} & {\square}\end{bmatrix}\begin{bmatrix}{24} & {} & {} \\ {\square} & {} & {} \\ {\square} & {} & {}\end{bmatrix}\begin{cases}-2 \\ \square \\ \square\end{cases}$

Now we lower the first coefficient and multiply it by -2 and add that to the second coefficient:

$\begin{bmatrix}{1} & {-5} & {-2} \\ {\square} & {-2} & {\square} \\ {1} & {-7} & {\square}\end{bmatrix}\begin{bmatrix}{24} & {} & {} \\ {\square} & {} & {} \\ {\square} & {} & {}\end{bmatrix}\begin{cases}-2 \\ \square \\ \square\end{cases}$

Now we repeat the previous step. We multiply -7 by -2 and add that to the next coefficient:

$\begin{bmatrix}{1} & {-5} & {-2} \\ {\square} & {-2} & {14} \\ {1} & {-7} & {12}\end{bmatrix}\begin{bmatrix}{24} & {} & {} \\ {\square} & {} & {} \\ {\square} & {} & {}\end{bmatrix}\begin{cases}-2 \\ \square \\ \square\end{cases}$

Now we repeat the previous step. we multiply 12 by -2 and add that to the next coefficient:

$\begin{bmatrix}{1} & {-5} & {-2} \\ {\square} & {-2} & {14} \\ {1} & {-7} & {12}\end{bmatrix}\begin{bmatrix}{24} & {} & {} \\ {-24} & {} & {} \\ {0} & {} & {}\end{bmatrix}\begin{cases}-2 \\ \square \\ \square\end{cases}$

The last number we got is the residue of the division, in this case, it is 0. Now we rewrite the polynomial but we subtract 1 to the order of the polynomial:

$\frac{x^3-5x^2-2x+24}{x+2}=x^2-7x+12$