# How do the coefficients of a polynomial affects its end behavior?

Aug 28, 2015

For even degree polynomials, a positive leading coefficient implies $y \to + \infty$ as $x \to \pm \infty$, while a negative leading coefficient implies $y \to - \infty$ as $x \to \pm \infty$. For odd degree polynomials, a positive leading coefficient implies $y \to + \infty$ as $x \to + \infty$ and $y \to - \infty$ as $x \to - \infty$, while a negative leading coefficient implies $y \to - \infty$ as $x \to + \infty$ and $y \to + \infty$ as $x \to - \infty$.

#### Explanation:

A (real) polynomial of (integer) degree $n$ is a function of the form $p \left(x\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + {a}_{n - 2} {x}^{n - 2} + \cdots + {a}_{2} {x}^{2} + {a}_{1} x + {a}_{0}$, where ${a}_{n} \ne 0$ (otherwise it wouldn't be degree $n$), and all the other $a$'s are arbitrary real numbers (and they can be zero).

If $n$ is even, then ${a}_{n} > 0$ implies that $y \to + \infty$ as $x \to \pm \infty$ and ${a}_{n} < 0$ implies $y \to - \infty$ as $x \to \pm \infty$.

If $n$ is odd, then ${a}_{n} > 0$ implies that $y \to + \infty$ as $x \to + \infty$ and $y \to - \infty$ as $x \to - \infty$ and ${a}_{n} < 0$ implies that $y \to - \infty$ as $x \to + \infty$ and $y \to + \infty$ as $x \to - \infty$.

The values of the other coefficients are irrelevant for determining the end behavior.