Question

use the long division method to find the result when $2x^3 + 7x^2 + 19x + 8$ is divided by $2x + 1$.

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(2x^{3}\) by the leading term of the divisor \(2x\). We get \(x^{2}\). Multiply the divisor \(2x + 1\) by \(x^{2}\) to get \(2x^{3}+x^{2}\). Subtract this from the dividend:
\[

$$\begin{align*} (2x^{3}+7x^{2}+19x + 8)-(2x^{3}+x^{2})&=2x^{3}+7x^{2}+19x + 8 - 2x^{3}-x^{2}\\ &=6x^{2}+19x + 8 \end{align*}$$

\]

Step2: Divide the new leading terms

Divide the leading term of the new dividend \(6x^{2}\) by the leading term of the divisor \(2x\). We get \(3x\). Multiply the divisor \(2x + 1\) by \(3x\) to get \(6x^{2}+3x\). Subtract this from the new dividend:
\[

$$\begin{align*} (6x^{2}+19x + 8)-(6x^{2}+3x)&=6x^{2}+19x + 8 - 6x^{2}-3x\\ &=16x + 8 \end{align*}$$

\]

Step3: Divide the new leading terms

Divide the leading term of the new dividend \(16x\) by the leading term of the divisor \(2x\). We get \(8\). Multiply the divisor \(2x + 1\) by \(8\) to get \(16x+8\). Subtract this from the new dividend:
\[

$$\begin{align*} (16x + 8)-(16x+8)&=16x + 8 - 16x - 8\\ &=0 \end{align*}$$

\]

Answer:

The result of the long - division of \(2x^{3}+7x^{2}+19x + 8\) by \(2x + 1\) is \(x^{2}+3x + 8\)