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{x^4 - 2x^3 + x^2 - 2x + 1}{x - 2}\\)
\\(r(x) = ?\\)
only enter the r(x) term.

Explanation:

Step1: Use Polynomial Division or Remainder Theorem

The Remainder Theorem states that when a polynomial \( f(x) \) is divided by \( x - a \), the remainder \( R \) is \( f(a) \). Here, we are dividing by \( x - 2 \), so \( a = 2 \). We need to find \( f(2) \) where \( f(x)=x^{4}-2x^{3}+x^{2}-2x + 1 \).

Step2: Substitute \( x = 2 \) into \( f(x) \)

\[

$$\begin{align*} f(2)&=(2)^{4}-2\times(2)^{3}+(2)^{2}-2\times(2)+1\\ &=16-2\times8 + 4-4 + 1\\ &=16-16+4 - 4+1\\ &=1 \end{align*}$$

\]
Since the divisor is linear (\( x - 2 \)), the remainder \( R(x) \) is a constant (degree 0 polynomial), so \( R(x)=1 \).

Answer:

\( 1 \)