Question

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. f((x) = (x + 1)(x + 4)(x + 5)^5

Answer 1

The graph falls to the left and rises to the right.

Explanation:

Given f(x) defined below:

`f\mleft(x)=(x+1)(x+4)(x+5)^5\mright?`

We are to determine the end behavior of the polynomial using the Leading Coefficient Test.

When using the Leading coefficient test, the following rule applies:

• When the ,degree is odd, and the ,leading coefficient is positive,, the graph falls to the left and rises to the right.

,

• When the ,degree is odd, and the ,leading coefficient is negative,, the graph rises to the left and falls to the right.

,

• When the ,degree is even, and the ,leading coefficient is positive,, the graph rises to the left and right.

,

• When the ,degree is even, and the ,leading coefficient is negative,, the graph falls to the left and right.

Back to our function, f(x):

`\begin{gathered} f(x)=(x+1)(x+4)(x+5)^5 \\ \text{Degree}=7\text{ (Odd)} \\ \text{Leading Coefficient = 1 (Positive)} \end{gathered}`

From the first rule above, we can conclude that the graph falls to the left and rises to the right.

A graph of f(x) is attached which confirms this end behavior.