question\nfind the graph of the polynomial given below.\n\nselect the correct answer below:

Answer:

First option (the first graph)

Explanation:

Step1: Find the roots of the polynomial

Set (f(x)=0), then (2(x - 5)(x + 1)(x + 8)=0). Using the zero - product property (a\times b\times c = 0\Rightarrow a = 0) or (b = 0) or (c = 0). We get (x=5), (x=-1), (x=-8).

Step2: Determine the end - behavior

The degree of the polynomial (y = 2(x - 5)(x + 1)(x + 8)=2x^{3}+8x^{2}-78x - 80) is (n = 3) (odd) and the leading coefficient (a = 2>0). As (x\rightarrow-\infty), (y=2x^{3}(1+\frac{4}{x}-\frac{39}{x^{2}}-\frac{40}{x^{3}})\rightarrow-\infty) (since (x^{3}\rightarrow-\infty) as (x\rightarrow-\infty) and (2>0)). As (x\rightarrow+\infty), (y = 2x^{3}(1+\frac{4}{x}-\frac{39}{x^{2}}-\frac{40}{x^{3}})\rightarrow+\infty) (since (x^{3}\rightarrow+\infty) as (x\rightarrow+\infty) and (2>0)).

The first graph has (x) - intercepts at (x=-8), (x = - 1), (x = 5) and the end - behavior (y\rightarrow-\infty) as (x\rightarrow-\infty) and (y\rightarrow+\infty) as (x\rightarrow+\infty).