Question

describe the end behavior, state the degree and leading coefficient of the polynomial. if the function is not a polynomial, explain why. (g(x)=8x^{4}+5x^{5})

a) as (x\to-infty), (g(x)\to-infty) and as (x\toinfty), (g(x)\to-infty). degree = 5; leading coefficient = 5

b) as (x\to-infty), (g(x)\toinfty) and as (x\toinfty), (g(x)\toinfty). degree = 5; leading coefficient = 5

c) as (x\to-infty), (g(x)\to-infty) and as (x\toinfty), (g(x)\toinfty). degree = 5; leading coefficient = 5

d) as (x\to-infty), (g(x)\to-infty) and as (x\toinfty), (g(x)\toinfty). degree = 4; leading coefficient = 8

Explanation:

Step1: Identify the degree of the polynomial

The degree of a polynomial is the highest - power of the variable. In the polynomial \(g(x)=8x^{4}+5x^{5}\), the highest power of \(x\) is 5, so the degree \(n = 5\).

Step2: Identify the leading coefficient

The leading coefficient is the coefficient of the term with the highest - power of the variable. For the polynomial \(g(x)=8x^{4}+5x^{5}\), the coefficient of \(x^{5}\) is 5, so the leading coefficient \(a = 5\).

Step3: Determine the end - behavior

For a polynomial \(y = a x^{n}\), when \(n\) is odd and \(a>0\):
As \(x\to-\infty\), \(y=a x^{n}\to-\infty\) (since \(x^{n}\to-\infty\) when \(n\) is odd and \(x\to-\infty\) and \(a>0\)), and as \(x\to\infty\), \(y = a x^{n}\to\infty\) (since \(x^{n}\to\infty\) when \(n\) is odd and \(x\to\infty\) and \(a>0\)).

Answer:

B. As \(x\to-\infty\), \(g(x)\to-\infty\) and as \(x\to\infty\), \(g(x)\to\infty\), degree = 5; leading coefficient = 5