# If you have a polynomial where the leading coefficient is positive and the degree is odd, what is the end behavior?

Feb 17, 2017

As $x \to - \infty$, $y \to - \infty$ and as $x \to \infty$, $y \to \infty$ and curve cuts $x$-axis at the zeros of the function $y = f \left(x\right)$

#### Explanation:

In a polynomial say $y = f \left(x\right)$, ($f \left(x\right)$ being a polynomial), where the leading coefficient is positive and the degree is odd,

this means that as $x \to - \infty$, as the degree is odd, $y \to - \infty$

and as $x \to \infty$, as the degree is odd, $y \to \infty$

Further in between the curve will cut the $x$-axis at all the zeros of the function $y = f \left(x\right)$

For example let $y = {\left(x - 2\right)}^{2} \left(x + 3\right) \left(x - 5\right) \left(x + 4\right)$

Here we observe the above behavior and it cuts $x$-axis at $\left\{- 4 , - 3 , 2 , 5\right\}$
graph{(x-2)^2(x+3)(x-5)(x+4) [-10, 10, -360, 300]}