Question

Consider the polynomial function. f(x) = x^2 (x−2) (x+5)^3

Which statements correctly describes the behavior of the graph of the function at its zeros?

The graph of the function crosses through the x-axis at −5 and behaves in a cubic fashion. The graph of the function touches and bounces off of the x-axis at 0 and crosses straight through the x-axis at 2.

The graph of the function crosses through the x-axis at 5 and behaves in a cubic fashion. The graph of the function touches and bounces off of the x-axis at 0 and crosses straight through the x-axis at −2.

The graph of the function crosses through the x-axis at 0 and behaves in a cubic fashion. The graph of the function touches and bounces off of the x-axis at 5 and crosses straight through the x-axis at −2.

The graph of the function crosses through the x-axis at −5 and behaves in a cubic fashion. The graph of the function touches and bounces off of the x-axis at 2 and crosses straight through the x-axis at 0.

Answer 1

• A. The graph of the function crosses through the x-axis at −5 and behaves in a cubic fashion. The graph of the function touches and bounces off of the x-axis at 0 and crosses straight through the x-axis at 2.

Explanation:

Given function:

• f(x) = x² (x - 2) (x + 5)³

It has zero's at:

• x = 0
• x - 2 = 0 ⇒ x = 2
• x + 5 = 0 ⇒ x = - 5

Observations:

• Two of zero's are repeat, x = 0 is twice and x = - 5 three times.
• Since it has 3 unique zero's, the function behaves in a cubic fashion.
• Since it has double zero's at 0, it only touches and bounces off at this point.
• Since it has single zero's at - 5 and 2, it crosses the x-axis at same points.

Considering all the mentioned above the matching answer choice is A