Question

A polynomial function has roots at –6 (with multiplicity 1), –2 (with multiplicity 3), 0 (with multiplicity 2), and 4 (with multiplicity 3). If the function has a positive leading coefficient and is of odd degree, which statement about the graph is true?

a. The graph is positive on (–6, –2)
b. The graph is negative on (-[infinity], 0)
c. The graph is positive on (–2, 4)
d. The graph is negative on (4, [infinity])

Answer 1

The graph is positive on (-6, -2).

Step-by-step explanation:

The polynomial function with the given roots is expressed as:

f(x) = a(x + 6)(x + 2)^3(x)(x - 4)^3

Since the function has a positive leading coefficient and is of odd degree, the graph will start in the third quadrant and end in the first quadrant. Therefore, the graph is positive on the intervals (-6, -2) and (0, 4).

So, the correct statement is: a. The graph is positive on (-6, -2).