# What is the end behavior of f(x) = 3x^4 - x^3 + 2x^2 + 4x + 5?

Sep 16, 2014

To find the end behavior you have to consider 2 items.

The first item to consider is the degree of the polynomial. The degree is determined by the highest exponent. In this example the degree is even , $4$.

Because the degree is even the end behaviors could be both ends extending to positive infinity or both ends extending to negative infinity.

The second item determines if those end behaviors are negative or positive. We now look at the coefficient of the term with the highest degree. In this example the coefficient is a positive $3$.

If that coefficient is positive then the end behaviors are positive.
If the coefficient is negative then the end behaviors are negative.

In this example the end behaviors are $\uparrow$ and $\uparrow$.

End behaviors:

Even degree and positive coefficient: $\uparrow$ and $\uparrow$

Even degree and negative coefficient: $\downarrow$ and $\downarrow$

Odd degree and positive coefficient: $\downarrow$ and $\uparrow$

Odd degree and negative coefficient: $\uparrow$ and $\downarrow$