a polynomial function has a root of -5 with multiplicity 3, a root of 1 with multiplicity 2, and a root of 3…

Answer:

The graph of the function is negative on $(3,\infty)$.

Explanation:

Step1: Determine the degree of the polynomial

The degree is $3 + 2+7=12$ (sum of multiplicities).

Step2: Analyze end - behavior

Since the leading coefficient $a<0$ and degree $n = 12$ (even), as $x\to\pm\infty$, $y\to-\infty$.

Step3: Consider the roots

The roots are $x=-5$ (multiplicity 3), $x = 1$ (multiplicity 2), $x = 3$ (multiplicity 7). The graph crosses the $x$ - axis at odd - multiplicity roots ($x=-5$ and $x = 3$) and touches at even - multiplicity root ($x = 1$).

Step4: Test intervals

For $x>3$, since the end - behavior as $x\to\infty$ is $y\to-\infty$, the graph is negative on $(3,\infty)$.