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 .

### Multiplying

Multiplying polynomials is based on the familiar property of
arithmetic, **distribution**: .

**binomials**(polynomials with only two terms), this is frequently referred to as

**FOIL**.

There are several product formulas that arise repeatedly when working with binomials. You will likely have seen most of these before.

### Factoring

Factoring is a bit like an inverse operation of multiplying polynomials. We start with the multiplied out polynomial, and ask what the individual factors were.

The easiest factors to deal with are common factors.

For **trinomials** (polynomials with three terms) of the form , we try to factor as .
We’ll start with an example with .

Our factorization is .

The process is slightly more complicated when .

All of these possibilities will give the right term and the right constant term. The only difference is the -term they give. Do any of them give an -coefficient of ? Yes!

.

Remember, a **root** of a polynomial function is an -value where the polynomial is zero.
There is a close relationship between roots of a polynomial and factors of that
polynomial. Precisely:

The quotient was , so . It remains to factor the quotient.

The leading coefficient factors as , and factors as and . Examining the possibilities, we find the factorization of as . The entire polynomial factors as .

### Equations

A *quadratic equation in * is an equation which is equivalent to one with the form ,
where , , and are constants, with .

There are three major techniques you are probably familiar with for solving quadratic equations:

- Factoring.
- Completing the Square.
- Quadratic Formula.

Each of these methods are important. Factoring is vital, because it is a valid approach to solve nearly any type of equation. Completing the Square is a technique that becomes useful when we need to rewrite certain types of expressions. The Quadratic Formula will always work, but has some limitations.

. Any of the above methods will work here, so let’s try factoring. What numbers add to and multiply to ? and do. That gives us:

Either (giving us ) or (giving us ). The two solutions are .

Building on these methods, we’ll consider more general types of equations. A polynomial equation is an equation which is equivalent to one of the form .

Factoring is a process that helps us solve more than just quadratic equations, as long as we first get one side of the equation equal to zero.

The next theorem above is a deep fact of mathematics. The great mathematician Gauss proved the theorem in 1799.

Recall that the **multiplicity** of a root indicates how many times that particular root
is repeated. The Fundamental Theorem of Algebra tells us that a polynomial
equation of degree , will have exactly complex solutions, once multiplicity is taken
into account. As non-real solutions appear in complex-conjugate pairs (if our
equation has real coefficients), we will always have an even number of non-real
solutions. That means, taking multiplicity into account, a quadratic equation will
have either or real solutions. A cubic equation will have either or real
solution.