Question

question
use the long division method to find the result when $8x^3 + 28x^2 + 14x - 13$ is divided by $2x + 3$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.
answer attempt 1 out of 2

Explanation:

Step 1: Divide the leading terms

Divide the leading term of the dividend \(8x^3\) by the leading term of the divisor \(2x\). So, \(\frac{8x^3}{2x} = 4x^2\). Multiply the divisor \(2x + 3\) by \(4x^2\) to get \(8x^3+12x^2\). Subtract this from the dividend:

$$
LATEXBLOCK0
$$

Step 2: Divide the new leading terms

Divide the leading term of the new dividend \(16x^2\) by the leading term of the divisor \(2x\). So, \(\frac{16x^2}{2x}=8x\). Multiply the divisor \(2x + 3\) by \(8x\) to get \(16x^2 + 24x\). Subtract this from the new dividend:

$$
LATEXBLOCK1
$$

Step 3: Divide the new leading terms

Divide the leading term of the new dividend \(-10x\) by the leading term of the divisor \(2x\). So, \(\frac{-10x}{2x}=-5\). Multiply the divisor \(2x + 3\) by \(-5\) to get \(-10x-15\). Subtract this from the new dividend:

$$
LATEXBLOCK2
$$

The quotient \(q(x)=4x^2 + 8x-5\) and the remainder \(r(x) = 2\) and the divisor \(b(x)=2x + 3\). So the result is \(4x^2+8x - 5+\frac{2}{2x + 3}\)

Answer:

\(4x^2 + 8x - 5+\frac{2}{2x + 3}\)