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^3 + 8x^2 + 12x - 7}{x - 5}\\)
\\(r(x) = ?\\)
only enter the r(x) term.

Explanation:

Step1: Recall Polynomial Division Remainder

When dividing a polynomial \( f(x) \) by \( d(x) = x - a \), the remainder \( R(x) \) is a constant (since the degree of \( d(x) \) is 1, the remainder has degree 0) and can be found using the Remainder Theorem: \( R = f(a) \). Here, \( d(x)=x - 5 \), so \( a = 5 \).

Step2: Substitute \( x = 5 \) into \( f(x) \)

Given \( f(x)=-2x^{3}+8x^{2}+12x - 7 \), substitute \( x = 5 \):
\[

$$\begin{align*} f(5)&=-2(5)^{3}+8(5)^{2}+12(5)-7\\ &=-2(125)+8(25)+60 - 7\\ &=-250 + 200+60 - 7\\ &=(-250 + 200)+(60 - 7)\\ &=-50 + 53\\ &= 3 \end{align*}$$

\]
Since the remainder is a constant (degree 0 polynomial), \( R(x)=3 \).

Answer:

\( 3 \)