A polynomial function has a root of –7 with multiplicity 2, a root of –1 with multiplicity 1, a root of 2 with multiplicity 4, and a root of 4 with multiplicity 1. If the function has a positive leading coefficient and is of even degree, which statement about the graph is true? The graph of the function is positive on (2, 4). The graph of the function is negative on (4, infinity). The graph of the function is positive on (-infinity, –7). The graph of the function is negative on (–7, –1).

C) 840,000a

From the information given in the problem, our function will be

f(x) = a(x + 7)^2 (x + 1)(x - 2)^4 (x - 4)

where a is a positive leading coefficient.

Now, we just need to go through the answer choices and plug in x values that are in each interval.

A) 3 is between (2,4) so let's test f(3)

f(3) = -400a

-400a is not a positive number, so A is wrong

B) f(5) = 69984a

69984a is not a negative number, so B is also wrong.

C) f(-8) = 840,000a
840,000a is a positive number, so C is correct.
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D) Just to make sure, let's check D.
f(-2) = 38400a

38400a is not a negative number, so D is also wrong.