Jessie is stuck on her math homework. She is stuck on the problem

. The directions are asking her to simplify, but she isn’t sure how to do that. Do you know how to help her?In this concept, you will learn to simplify polynomials by combining like terms.

### Combining Like Terms

A **polynomial** is an algebraic expression that shows the sum of monomials. Since the prefix mono means one, a monomial is a single piece or term. The prefix poly means many. So the word polynomial refers to one or more than one term in an expression. The relationships between these terms may be sums or differences.

Polynomial expressions include: \begin{align*}x^2+5 \qquad 3x-8+4x^5 \qquad -7a^2+9b-4b^3+6\end{align*}

You can simplify polynomials by combining like terms. In mathematics, you are able to combine like terms but you cannot combine unlike terms.

Terms are considered **like terms** if they have exactly the same variables with exactly the same exponents.

A term can also be a single number like 7 or -5. These are called **constants**.

Any term with a variable has a numerical factor called the **coefficient**. The coefficient of is 4. The coefficient of \begin{align*}-7a^2\end{align*}

Let’s look at some examples of like and unlike terms.

\begin{align*}n\end{align*}

\begin{align*}4n^2\end{align*}

\begin{align*}8y^3\end{align*}

Like terms can be combined by adding their coefficients.

Notice that the exponent does not change when you combine like terms. If you think of \begin{align*}n + n + n + n + n + n + n\end{align*} and as a shorter way of writing , then combining those like terms to get is a simpler way to write \begin{align*}7n + 5n\end{align*}.

as simply a shorter way of writing### Examples

#### Example 1

Earlier, you were asked about helping Jessie with her simplification problem.

Here is the problem that Jessie is stuck on:

\begin{align*}5x - 3y - 9x + 7y\end{align*}

First, look at the like terms and combine them:

\begin{align*}\begin{array}{rcl} 5x - 9x &=& -4x \\ -3y + 7y &=& 4y \end{array}\end{align*}

Then write the terms in a single expression again:

The answer is \begin{align*}-4x + 4y\end{align*}.

#### Example 2

Simplify by combining like terms:

First, let’s look at the like terms and combine them.

Then rewrite the combined terms in a single expression:

The answer is \begin{align*}-5x+10y+4\end{align*}.

#### Example 3

Simplify by combining like terms.

First, let’s look at the like terms and combine them.

Then rewrite the terms in a single expression:

The answer is

.#### Example 4

Simplify by combining like terms.

First, let’s look at the like terms and combine them.

\begin{align*}\begin{array}{rcl} 5a + a &=& 6a \\ 3b - 8b &=& -5b \\ -7 \end{array}\end{align*}

Then rewrite the terms in a single expression:

The answer is

.#### Example 5

Simplify by combining like terms.

First, look at the like terms and combine them.

\begin{align*}\begin{array}{rcl}
5a - 2a + 8a &=& 11a \\
-7b + 8b &=& b \\
-9 + 8 &=& -1
\end{array}\end{align*}

Then rewrite the combined terms in a single expression:

The answer is

.### Review

Simplify the following polynomials by combining like terms.

- \begin{align*}5x - 7x + 5x + 4 -9\end{align*}
- \begin{align*}9x + 7y - 15x + 4x - 9y\end{align*}