Polynomials are some of our favorite functions.

### What are polynomial functions?

**polynomial function**in the variable is a function which can be written in the form where the ’s are all constants (called the

**coefficients**) and is a whole number (called the

**degree**when ). The domain of a polynomial function is .

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.

**even**or

**odd**degree, and if the leading coefficient (the one next to the highest power of ) of the polynomial is

**positive**or

**negative**.

- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.
- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.
- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.
- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.