Solving Polynomial Remainder Problem

This solution explains how to find the remainder when the polynomial $P(x) = x^4 + x^3 - x^2 + x + 1$ is divided by the binomial $x - 3$. We will use the Remainder Theorem.

Remainder Theorem Application

The Remainder Theorem states that if a polynomial $P(x)$ is divided by a linear divisor of the form $x - c$, the remainder is equal to the value of the polynomial evaluated at $x = c$, which is $P(c)$.

Step-by-Step Calculation

In this problem, the polynomial is $P(x) = x^4 + x^3 - x^2 + x + 1$ and the divisor is $x - 3$. Here, $c = 3$.

1. Identify the value of $c$ from the divisor $x - c$. In this case, $c = 3$.
2. Substitute $x = 3$ into the polynomial $P(x)$:

   $P(3) = (3)^4 + (3)^3 - (3)^2 + (3) + 1$

3. Calculate the powers of 3:

   $P(3) = 81 + 27 - 9 + 3 + 1$

4. Perform the addition and subtraction:

   $P(3) = 108 - 9 + 4$

   $P(3) = 99 + 4$

   $P(3) = 103$

Therefore, the remainder when $x^4 + x^3 - x^2 + x + 1$ is divided by $x - 3$ is 103.