Question

divide.
\\((x^5 + 1) \div (x + 1)\\)

the quotient is \\(\square\\) with a remainder of \\(\square\\).
(simplify your answers. use integers or fractions for any num

Explanation:

Step1: Use sum of 5th powers identity

Recall $a^5 + b^5 = (a+b)(a^4 - a^3b + a^2b^2 - ab^3 + b^5)$. Here $a=x$, $b=1$:
$$x^5 + 1 = (x+1)(x^4 - x^3 + x^2 - x + 1)$$

Step2: Divide by $(x+1)$

Cancel the common factor $(x+1)$:
$$\frac{x^5 + 1}{x+1} = x^4 - x^3 + x^2 - x + 1 + \frac{0}{x+1}$$

Answer:

The quotient is $x^4 - x^3 + x^2 - x + 1$ with a remainder of $0$.