Question

divide.
\\((x^5 - 32) \div (x - 2)\\)

the quotient is \\(\square\\) with a remainder of \\(\square\\).
(simplify your answers. use integers or fractions for any numbers in the expression. do not factor.)

Explanation:

Step1: Recognize difference of 5th powers

$x^5 - 32 = x^5 - 2^5$

Step2: Apply difference of powers formula

Recall $a^n - b^n = (a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})$
For $n=5$, $a=x$, $b=2$:
$$x^5 - 2^5 = (x-2)(x^4 + 2x^3 + 4x^2 + 8x + 16)$$

Step3: Perform the division

$\frac{x^5 - 32}{x-2} = \frac{(x-2)(x^4 + 2x^3 + 4x^2 + 8x + 16)}{x-2}$
Cancel $(x-2)$ (where $x
eq2$):
$\frac{x^5 - 32}{x-2} = x^4 + 2x^3 + 4x^2 + 8x + 16$

Step4: Identify quotient and remainder

The division results in no remainder, so remainder is 0.

Answer:

The quotient is $x^4 + 2x^3 + 4x^2 + 8x + 16$ with a remainder of $0$.