Question

use synthetic division to find the result when $x^4 + 8x^3 + 24x^2 + 25x - 14$ is divided by $x + 3$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify the root for synthetic division

For the divisor \(x + 3\), the root is \(x=-3\) (since \(x + 3=0\) gives \(x=-3\)). The coefficients of the dividend \(x^{4}+8x^{3}+24x^{2}+25x - 14\) are \(1,8,24,25,-14\).

Step2: Set up synthetic division

We set up the synthetic division as follows:

$$
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$$

The last number \(-8\) is the remainder, and the other numbers \(1,5,9,-2\) are the coefficients of the quotient polynomial.

Step3: Write the quotient and remainder

The quotient polynomial \(q(x)\) is \(x^{3}+5x^{2}+9x - 2\) and the remainder \(r(x)=-8\), and the divisor \(b(x)=x + 3\). So the result is \(q(x)+\frac{r(x)}{b(x)}=x^{3}+5x^{2}+9x - 2+\frac{-8}{x + 3}\) or \(x^{3}+5x^{2}+9x - 2-\frac{8}{x + 3}\).

Answer:

\(x^{3}+5x^{2}+9x - 2-\frac{8}{x + 3}\)