Question

use synthetic division to find the result when $4x^3 + 14x^2 + 13x + 11$ 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: Identify the root of the divisor

For the divisor \(x + 2\), set \(x + 2 = 0\), so \(x=-2\). The coefficients of the dividend \(4x^{3}+14x^{2}+13x + 11\) are \(4\), \(14\), \(13\), \(11\).

Step2: Perform synthetic division

Bring down the first coefficient \(4\).
Multiply \(4\) by \(-2\) to get \(-8\), add to the next coefficient: \(14+(-8) = 6\).
Multiply \(6\) by \(-2\) to get \(-12\), add to the next coefficient: \(13+(-12)=1\).
Multiply \(1\) by \(-2\) to get \(-2\), add to the last coefficient: \(11+(-2) = 9\).
The coefficients of the quotient polynomial (starting from degree \(2\)) are \(4\), \(6\), \(1\), and the remainder is \(9\). So the quotient \(q(x)=4x^{2}+6x + 1\) and the remainder \(r(x) = 9\), and the divisor \(b(x)=x + 2\).

Answer:

\(4x^{2}+6x + 1+\frac{9}{x + 2}\)