A polynomial of odd degree has a real root

A polynomial of odd degree always has at least one real root. This means that if you have a polynomial like $f(x) = x^3 + 2x + 1$ (degree 3), there is some real number $r$ such that $f(r) = 0$. This happens because, as $x$ becomes very large or very small, the polynomial's value goes to $+ ext{infinity}$ in one direction and $- ext{infinity}$ in the other, so the graph must cross the $x$-axis somewhere.

Important properties

• The leading term (the term with the highest power) determines the end behavior of the polynomial.

• For odd degree polynomials, as $x \to +\infty$, $f(x) \to +\infty$ or $-\infty$ depending on the leading coefficient, and as $x \to -\infty$, $f(x)$ goes to the opposite infinity.

• Because the function changes sign, by the Intermediate Value Theorem (or by observing the graph), it must cross the $x$-axis at least once.