Question

andre is dividing $2x^{3}+15x^{2}+34x + 24$ by $x + 2$. after dividing into $2x^{3}$, andre writes $2x^{2}$ as the first term of the quotient and then multiplies $2x^{2}$ by the divisor. what is the next step in the long division?

a) subtract $(2x^{3}+4x^{2})$ from $(2x^{3}+15x^{2})$.

b) subtract $(2x^{3}-4x^{2})$ from $(2x^{3}+15x^{2})$.

c) add $(2x^{3}+4x^{2})$ to $(2x^{3}+15x^{2})$.

d) add $(-2x^{3}+4x^{2})$ to $(2x^{3}+15x^{2})$.

Explanation:

Step1: Recall Long Division Steps

In polynomial long division, after multiplying the quotient term by the divisor, we subtract the result from the corresponding part of the dividend. Here, the divisor is \(x + 2\) and the quotient term is \(2x^{2}\). Multiplying them gives \(2x^{2}(x + 2)=2x^{3}+4x^{2}\).

Step2: Determine the Next Operation

We need to subtract this product (\(2x^{3}+4x^{2}\)) from the part of the dividend involving the \(x^{3}\) and \(x^{2}\) terms, which is \(2x^{3}+15x^{2}\). So the next step is to subtract \((2x^{3}+4x^{2})\) from \((2x^{3}+15x^{2})\).

Answer:

A) Subtract \((2x^{3}+4x^{2})\) from \((2x^{3}+15x^{2})\).