Question

A polynomial function has roots at -5 with multiplicity 3, 1 with multiplicity 2, and 3 with multiplicity 7. The function has a negative leading coefficient and is of even degree. Which statement about the graph is true?

a) The graph of the function is positive on (-[infinity], -5).
b) The graph of the function is negative on (-5, 3).
c) The graph of the function is positive on (-[infinity], 1).
d) The graph of the function is negative on (1, 3).

Answer 1

The graph of the polynomial function is positive on (-∞, -5).

Step-by-step explanation:

The polynomial function described in the question has roots at -5 with multiplicity 3, 1 with multiplicity 2, and 3 with multiplicity 7. Since the function has a negative leading coefficient and is of even degree, it means that the graph of the function will be positive on the intervals where the multiplicity of the roots is odd, and negative on the intervals where the multiplicity of the roots is even.

This means that the graph of the function is positive on (-∞, -5) and (1, ∞), and negative on (-5, 1).

Therefore, the correct statement is a) The graph of the function is positive on (-∞, -5).