Question

question
use the long division method to find the result when $4x^3 + 4x^2 + 19x + 9$ is divided by $2x + 1$.

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(4x^{3}\) by the leading term of the divisor \(2x\). So, \(\frac{4x^{3}}{2x} = 2x^{2}\).
Multiply the divisor \(2x + 1\) by \(2x^{2}\): \(2x^{2}(2x + 1)=4x^{3}+2x^{2}\).
Subtract this from the dividend: \((4x^{3}+4x^{2}+19x + 9)-(4x^{3}+2x^{2}) = 2x^{2}+19x + 9\).

Step2: Divide the new leading terms

Divide the leading term of \(2x^{2}+19x + 9\) (which is \(2x^{2}\)) by the leading term of the divisor \(2x\). So, \(\frac{2x^{2}}{2x}=x\).
Multiply the divisor \(2x + 1\) by \(x\): \(x(2x + 1)=2x^{2}+x\).
Subtract this from \(2x^{2}+19x + 9\): \((2x^{2}+19x + 9)-(2x^{2}+x)=18x + 9\).

Step3: Divide the new leading terms

Divide the leading term of \(18x + 9\) (which is \(18x\)) by the leading term of the divisor \(2x\). So, \(\frac{18x}{2x}=9\).
Multiply the divisor \(2x + 1\) by \(9\): \(9(2x + 1)=18x + 9\).
Subtract this from \(18x + 9\): \((18x + 9)-(18x + 9)=0\).

Answer:

The result of the long division is \(2x^{2}+x + 9\)