Question

use the long division method to find the result when $3x^3 + 10x^2 + 11x + 24$ is divided by $x + 3$.

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(3x^{3}\) by the leading term of the divisor \(x\). So, \(\frac{3x^{3}}{x}=3x^{2}\). This is the first term of the quotient.
Multiply the divisor \(x + 3\) by \(3x^{2}\): \(3x^{2}(x + 3)=3x^{3}+9x^{2}\).
Subtract this from the dividend: \((3x^{3}+10x^{2}+11x + 24)-(3x^{3}+9x^{2})=x^{2}+11x + 24\).

Step2: Divide the new leading terms

Now, divide the leading term of the new dividend \(x^{2}\) by the leading term of the divisor \(x\). So, \(\frac{x^{2}}{x}=x\). This is the second term of the quotient.
Multiply the divisor \(x + 3\) by \(x\): \(x(x + 3)=x^{2}+3x\).
Subtract this from the new dividend: \((x^{2}+11x + 24)-(x^{2}+3x)=8x + 24\).

Step3: Divide the new leading terms again

Divide the leading term of the new dividend \(8x\) by the leading term of the divisor \(x\). So, \(\frac{8x}{x}=8\). This is the third term of the quotient.
Multiply the divisor \(x + 3\) by \(8\): \(8(x + 3)=8x+24\).
Subtract this from the new dividend: \((8x + 24)-(8x + 24)=0\).

Answer:

\(3x^{2}+x + 8\)