Question

use synthetic division to find the result when $4x^3 + 10x^2 + 13x + 9$ is divided by $x + 1$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify root of divisor

For $x+1=0$, root is $x=-1$.

Step2: Set up synthetic division

Write coefficients of dividend $4x^3+10x^2+13x+9$: $4, 10, 13, 9$, and root $-1$:

$$\begin{array}{r|rrrr} -1 & 4 & 10 & 13 & 9 \\ \hline & & -4 & -6 & -7 \\ \hline & 4 & 6 & 7 & 2 \end{array}$$

Step3: Derive quotient and remainder

The first three values are coefficients of quotient $q(x)$: $4x^2+6x+7$. Last value is remainder $r(x)=2$. Divisor $b(x)=x+1$.

Step4: Write final form

Combine into $q(x)+\frac{r(x)}{b(x)}$.

Answer:

$4x^2 + 6x + 7 + \frac{2}{x+1}$