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: Calculate the degree of the polynomial

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

Step2: Analyze the sign - changes at roots

A root $r$ with odd multiplicity causes the graph to cross the $x$ - axis at $x = r$, and a root with even multiplicity causes the graph to touch the $x$ - axis at $x = r$. The roots are $x=-5$ (multiplicity 3), $x = 1$ (multiplicity 2), and $x = 3$ (multiplicity 7). Since the leading coefficient is negative and the degree is even, as $x\to\pm\infty$, $y\to-\infty$.

Step3: Test intervals

For $x\in(-\infty,-5)$, pick $x=-6$. The polynomial is negative. For $x\in(-5,1)$, pick $x = 0$. The polynomial is positive. For $x\in(1,3)$, pick $x = 2$. The polynomial is positive. For $x\in(3,\infty)$, pick $x = 4$. The polynomial is negative.