
Polynomials are some of our favorite functions.

The functions you are most familiar with are probably polynomial functions.

### What are polynomial functions?

Which of the following are polynomial functions?
$$ $$ $$ $$ $$ $$ $$ $$

The phrase above “in the variable $$” can actually change. is a polynomial in $$, and is a polynomial in $$.

### What can the graphs look like?

Fun fact:

Remember, a root is where a polynomial is zero. The theorem above is a deep fact of mathematics. The great mathematician Gauss proved the theorem in 1799 for his doctoral thesis.

The upshot as far as we are concerned is that when we plot a polynomial of degree $$, its graph will cross the $$-axis at most $$ times.