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.