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)}\\)

\\(f(x) = 9x^2 - 3x + 15\\) \\(d(x) = x + 1\\)

\\(r(x) = ?\\)

only enter the r(x) term.

Explanation:

Step1: Use polynomial long division or remainder theorem

We can use the remainder theorem, which states that the remainder of dividing a polynomial \( f(x) \) by \( x - a \) is \( f(a) \). Here, \( d(x)=x + 1=x-(-1) \), so \( a=-1 \).

Step2: Calculate \( f(-1) \)

Substitute \( x = -1 \) into \( f(x)=9x^{2}-3x + 15 \):
\[

$$\begin{align*} f(-1)&=9(-1)^{2}-3(-1)+15\\ &=9(1)+3 + 15\\ &=9+3+15\\ &=27 \end{align*}$$

\]
Since the divisor is linear (\( x + 1 \)), the remainder \( R(x) \) is a constant, so \( R(x)=27 \).

Answer:

27