Question

use the long division method to find the result when $3x^3 + 11x^2 + 16x + 10$ is divided by $3x + 5$.

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(3x^{3}+11x^{2}+16x + 10\) (which is \(3x^{3}\)) by the leading term of the divisor \(3x + 5\) (which is \(3x\)). So, \(\frac{3x^{3}}{3x}=x^{2}\).
Multiply the divisor \(3x + 5\) by \(x^{2}\): \((3x + 5)\times x^{2}=3x^{3}+5x^{2}\).
Subtract this from the dividend: \((3x^{3}+11x^{2}+16x + 10)-(3x^{3}+5x^{2}) = 6x^{2}+16x + 10\).

Step2: Divide the new leading terms

Divide the leading term of the new dividend \(6x^{2}+16x + 10\) (which is \(6x^{2}\)) by the leading term of the divisor \(3x\). So, \(\frac{6x^{2}}{3x}=2x\).
Multiply the divisor \(3x + 5\) by \(2x\): \((3x + 5)\times2x = 6x^{2}+10x\).
Subtract this from the new dividend: \((6x^{2}+16x + 10)-(6x^{2}+10x)=6x + 10\).

Step3: Divide the new leading terms again

Divide the leading term of the new dividend \(6x + 10\) (which is \(6x\)) by the leading term of the divisor \(3x\). So, \(\frac{6x}{3x}=2\).
Multiply the divisor \(3x + 5\) by \(2\): \((3x + 5)\times2=6x + 10\).
Subtract this from the new dividend: \((6x + 10)-(6x + 10)=0\).

Answer:

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