Question

which statement best describes the zeros of the function $h(x) = (x - 6)(x^2 + 8x + 16)$?

a. the function has two distinct real zeros.

b. the function has three distinct real zeros.

c. the function has one real zero and two complex zeros.

d. the function has three complex zeros.

Explanation:

Step1: Find zeros from first factor

Set \(x - 6 = 0\), so \(x = 6\) (a real zero).

Step2: Factor the quadratic

Factor \(x^2 + 8x + 16\). It's a perfect square: \((x + 4)^2\).

Step3: Find zeros from quadratic

Set \((x + 4)^2 = 0\), so \(x = - 4\) (a repeated real zero).

Step4: Analyze distinct real zeros

We have two distinct real zeros: \(x = 6\) and \(x = - 4\) (with multiplicity 2 for \(x=-4\)). So the function has two distinct real zeros.

Answer:

A. The function has two distinct real zeros.