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 \(x=-2\). We will use this value for synthetic division. The coefficients of the dividend \(4x^{3}+14x^{2}+13x + 11\) are \(4\), \(14\), \(13\), and \(11\).

Step2: Set up synthetic division

Write the root \(-2\) on the left and the coefficients \(4\), \(14\), \(13\), \(11\) in a row:

$$
LATEXBLOCK0
$$

Step3: Perform synthetic division

Bring down the first coefficient (\(4\)):

$$
LATEXBLOCK1
$$

Multiply the root \(-2\) by the brought - down value \(4\): \(-2\times4=-8\). Add this to the next coefficient (\(14\)): \(14+( - 8)=6\).

$$
LATEXBLOCK2
$$

Multiply the root \(-2\) by the new value \(6\): \(-2\times6 = - 12\). Add this to the next coefficient (\(13\)): \(13+( - 12)=1\).

$$
LATEXBLOCK3
$$

Multiply the root \(-2\) by the new value \(1\): \(-2\times1=-2\). Add this to the last coefficient (\(11\)): \(11+( - 2)=9\).

Step4: Interpret the results

The numbers in the bottom row (excluding the last one) are the coefficients of the quotient polynomial, and the last number is the remainder. The quotient polynomial \(q(x)\) has degree \(3 - 1=2\) (since we divided a cubic polynomial by a linear polynomial). So the quotient \(q(x)=4x^{2}+6x + 1\) and the remainder \(r(x)=9\), and the divisor \(b(x)=x + 2\).

Answer:

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