Question

Use the remainder theorem to find the remainder for (x ^5 + 32) divided by (x+2) and state whether or not the binomial is a factor of the polynomial A:0; yes B: 0;no C: 1; yes D: -1; no

Answer 1

ANSWER

The correct answer is A

EXPLANATION

The given polynomial is

$p(x) = {x}^{5} + 32$

According to the remainder theorem, if p(x) is divided by x+2 the remainder is p(-2).

If the remainder is zero then x+2 is a factor of f(x).

We plug in x=-2 into the function to obtain;

$p( - 2) = { (- 2)}^{5} + 32$

$p( - 2) = - 32 + 32$

$p( - 2) = 0$

Since the remainder is zero, x+2 is a factor.

The correct answer is A

Answer 2

$x+2$ is a factor of $x^5+32$ because (by the remainder theorem) the remainder upon dividing $x^5+32$ by $x+2$ is $(-2)^5+32=0$.

There's also the sum of fifth powers formula,

$a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$