Question

divide f(x) by d(x). your answer should be in the following format:
\\(\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}\\)
\\(\frac{f(x)}{d(x)} = \frac{2x^4 + 3x^3 - 16x^2 - 3x + 7}{2x - 1}\\)
\\(r(x) = ?\\)
only report the r(x) term.

Explanation:

Step1: Use Polynomial Long Division

We divide \(2x^{4}+3x^{3}-16x^{2}-3x + 7\) by \(2x - 1\).
First term: \(\frac{2x^{4}}{2x}=x^{3}\). Multiply \(2x - 1\) by \(x^{3}\): \(2x^{4}-x^{3}\). Subtract from the dividend: \((2x^{4}+3x^{3}-16x^{2}-3x + 7)-(2x^{4}-x^{3}) = 4x^{3}-16x^{2}-3x + 7\).

Step2: Next Term

\(\frac{4x^{3}}{2x}=2x^{2}\). Multiply \(2x - 1\) by \(2x^{2}\): \(4x^{3}-2x^{2}\). Subtract: \((4x^{3}-16x^{2}-3x + 7)-(4x^{3}-2x^{2})=-14x^{2}-3x + 7\).

Step3: Next Term

\(\frac{-14x^{2}}{2x}=-7x\). Multiply \(2x - 1\) by \(-7x\): \(-14x^{2}+7x\). Subtract: \((-14x^{2}-3x + 7)-(-14x^{2}+7x)=-10x + 7\).

Step4: Next Term

\(\frac{-10x}{2x}=-5\). Multiply \(2x - 1\) by \(-5\): \(-10x + 5\). Subtract: \((-10x + 7)-(-10x + 5)=2\).

Answer:

\(2\)