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

\\(r(x) = \boxed{?}x + \boxed{}\\)

only enter the r(x) term.

Explanation:

Step1: Perform polynomial long division

Divide \( f(x) = x^3 + 1 \) by \( d(x) = x^2 - x - 1 \).
First term of quotient: \( \frac{x^3}{x^2} = x \).
Multiply \( d(x) \) by \( x \): \( x(x^2 - x - 1) = x^3 - x^2 - x \).
Subtract from \( f(x) \): \( (x^3 + 1) - (x^3 - x^2 - x) = x^2 + x + 1 \).

Step2: Divide the new polynomial

Next term of quotient: \( \frac{x^2}{x^2} = 1 \).
Multiply \( d(x) \) by \( 1 \): \( 1(x^2 - x - 1) = x^2 - x - 1 \).
Subtract from \( x^2 + x + 1 \): \( (x^2 + x + 1) - (x^2 - x - 1) = 2x + 2 \).

So the remainder \( R(x) = 2x + 2 \).

Answer:

\( 2x + 2 \)