What is a Polynomial? | More Practice |

Multiplying Polynomials: FOILING, and “Pushing Through” |

## What is a Polynomial?

About the time when you learn exponents in variables, you’ll also learn how to add, subtract, multiply, and then divide (or factor) what we call **polynomials**. A polynomial (meaning “many” “nomial’s”, or many terms) is an expression that includes variables, constants, and exponents (not radicals) that are combined by addition, subtraction, and multiplication. They can also be combined by division, but no variables can be in the denominator; these are called **rational functions**, and we will work with them later in the **Rational Functions, Equations, and Inequalities** section. Note that in order for a function to be a **polynomial**, it’s** domain must be all real numbers**!

(Note that more multiplying (and factoring) polynomials will be in the **Introduction to Quadratics** and the **Graphing and Finding Roots of Polynomial Functions **sections.)

A polynomial with one term is called a **monomial**; two terms is a **binomial**; three terms is a **trinomial**, and four and more terms is typically just called a polynomial. There can be one or more variables in the polynomials, or no variables in polynomials (for example, if there is just a number, or **constant**).

The **degree** of the polynomial is the **highest exponent** of one of the terms (add exponents if there are more than one variable in that term); for example, the degree of $ {{x}^{3}}y$ is $ 3+1=4$.

A polynomial having a degree higher than five is typically just called a **polynomial**, or a **polynomial of degree **$ n$. Otherwise, a degree is five is called a **quintic, **a degree of four a **quartic**, three a **cubic**, two a **quadratic** (you’ll hear a lot more about these!), **one** a linear, and zero (just a number) a **constant**. Here are some examples:

These are **not** polynomials (notice the variable in the denominator or a root of a variable):

When adding and subtracting polynomials, put the terms with the **same variables and exponents together**. These are called “**like terms**”. For example, $ 3{{x}^{2}}y$ and $ -{{x}^{2}}y$ are like terms (adding them would be $ 2{{x}^{2}}y$), $ 4xy$ and $ yx$ are like terms (adding them would be $ 5xy$), but $ 4{{x}^{2}}{{y}^{2}}$ and $ 4x{{y}^{2}}$ are not.

We’re actually using the **distributive property** (sort of backwards) when we put together like terms. We saw this when **Solving Algebraic (Linear) Functions**, but it applies to other functions as well. Notice that we have invisible “$ 1$”’s before variables with no numbers (coefficients). For example:

## Multiplying Polynomials: FOILING, and “Pushing Through”

**Multiplying and dividing monomials** (one term) is straightforward. For multiplying, distribute the monomial, and for dividing by a monomial, break up the fraction into separate terms and simplify. Here are examples:

$ \require {cancel} \displaystyle \begin{array}{c}4y\left( {3{{y}^{2}}-4} \right)=12{{y}^{3}}-16y\,\,\,\,\,\,\,\,\,\\\displaystyle \frac{{12{{y}^{3}}-16y}}{{4y}}=\displaystyle \frac{{{{{\cancel{{12}}}}^{3}}{{y}^{3}}}}{{\cancel{4}y}}-\displaystyle \frac{{{{{\cancel{{16}}}}^{4}}\cancel{y}}}{{\cancel{4}\cancel{y}}}=3{{y}^{2}}-4\end{array}$

**Multiplying** **binomials** (also sometimes called **FOILING**) gets a little more complicated. There are different ways to do this: the **FOIL** (First Outer Inner Last) method or the more generic “pushing through” (distributing) method, or just doing “long multiplication”. Later, we’ll learn how to **undo** the multiplication of the polynomials (factoring) when we want to turn them back into factors.

You can use a **multiplication box** to multiply polynomials; let’s multiply $ \left( {x+3} \right)\left( {x-2} \right)$. Split up the first binomial on the top, and the second down the left side. (Watch the signs, and turn the **minus** $ 2$ in the second binomial into a $ -2$). Multiply across and down to fill in all the boxes, and then add up all the boxes:

Since we won’t want to draw a box every time we multiply binomials, we have a several methods to help us.

**FOILing**

**FOIL** stands for **First Outer Inner Last**. Multiply together the First, Outer, Inner, and then Last terms, and put plus signs between them (or a minus sign if the product is negative). **Note that FOILing only works if you multiply binomials – each factor has two terms. **In other cases, we’ll do the pure “**distributing**” method.

Note the last two examples are “special cases” that you’ll see a lot: **difference of two squares**, and **perfect square trinomials**; there are shortcuts for these cases.

**Notice how the middle term is always the sum of the products of the inside and outside terms; **soon this will become second nature and you won’t think about it!

**“Pushing Through” or Distributing Terms of Polynomials**

When we are **FOIL**ing, we are actually using the **Distributive Property** to make sure every term (variable or number) is multiplied by every other one and then you add them all up. We can also think of this as “pushing through” the terms to every other term. Also called “**double distributing**”, this way of multiplying binomials is more popular now, since it can be used for any polynomial.

Here are some examples; notice how this way is **much more generic and can be used with polynomials with any number of terms**.

**Learn these rules, and practice, practice, practice!**

**For Practice**: Use the **Mathway** widget below to try a **Polynomial Multiplication** problem. Click on **Submit** (the blue arrow to the right of the problem) and click on **Multiply **to see the answer**.**

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on “Tap to view steps”, or **Click Here**, you can register at **Mathway** for a **free trial**, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to **Introduction to Quadratics** – you are ready!