a polynomial function has a root of -6 with multiplicity 3 and a root of 2 with multiplicity 4. if the…

Answer:

The graph that has a zero - crossing at (x = - 6) (since multiplicity 3 is odd) and a touch - and - turn at (x = 2) (since multiplicity 4 is even), and falls to the right (because of negative leading coefficient and odd degree). Without seeing all options, we can analyze the general behavior. The function (f(x)=a(x + 6)^{3}(x - 2)^{4}), where (a<0). As (x\to-\infty), (y\to+\infty) and as (x\to+\infty), (y\to-\infty). The graph crosses the (x) - axis at (x=-6) and touches the (x) - axis at (x = 2).

Explanation:

Step1: Analyze multiplicity of roots

A root with odd multiplicity causes the graph to cross the (x) - axis. Here, root (x=-6) has multiplicity 3 (odd), so graph crosses at (x=-6). A root with even multiplicity causes the graph to touch and turn at the (x) - axis. Root (x = 2) has multiplicity 4 (even), so graph touches at (x = 2).

Step2: Analyze end - behavior

For a polynomial (y=a_nx^n+\cdots+a_0), if (n) (degree) is odd and (a_n<0) (negative leading coefficient), as (x\to-\infty), (y\to+\infty) and as (x\to+\infty), (y\to-\infty).