Question

use synthetic division to find the result when $2x^4 - 4x^3 - 13x^2 + 15x + 28$ is divided by $x + 2$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Adjust for synthetic division

For dividing by \(x + 2\), we use \(c=-2\) (since \(x+2=x - (-2)\)). The coefficients of the polynomial \(2x^{4}-4x^{3}-13x^{2}+15x + 28\) are \(2, - 4, - 13,15,28\).
Set up synthetic division:
\[

$$\begin{array}{r|rrrrr} -2 & 2 & -4 & -13 & 15 & 28\\ & & -4 & 16 & -6 & -18\\ \hline & 2 & -8 & 3 & 9 & 10\\ \end{array}$$

\]

Step2: Interpret the result

The coefficients of the quotient polynomial \(q(x)\) are \(2, - 8,3,9\) and the remainder \(r = 10\). The degree of the quotient is one less than the dividend (degree 4), so \(q(x)=2x^{3}-8x^{2}+3x + 9\) and the divisor \(b(x)=x + 2\). So the result is \(2x^{3}-8x^{2}+3x + 9+\frac{10}{x + 2}\).

Answer:

\(2x^{3}-8x^{2}+3x + 9+\frac{10}{x + 2}\)