# 12.3: Simplify Polynomials by Combining Like Terms

**At Grade**Created by: CK-12

**Practice**Simplify Expressions

Do you know how to identify like terms? Take a look at this dilemma.

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

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

The directions are asking her to simplify the problem, but she isn't sure how to do that.

Do you know?

**This Concept is all about combining like terms. You will know how to figure this out by the end of the Concept.**

### Guidance

**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 relationship between these terms may be sums or difference.

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

**We call an expression with a single term a** *monomial***, an expression with two terms is a** *binomial***, and an expression with three terms is a** *trinomial***. An expression with more than three terms is named simply by its number of terms—“five-term polynomial.”**

We can simplify polynomials by combining like terms. Take a look at this situation.

In a grocery store, a refrigerator in the back has 52 cartons of milk and 65 cans of soda. In the refrigerator near the cash registers, there are 12 cartons of milk and 26 cans of soda. How many do they have in all?

**Yes...there are 64 cartons of milk and 91 cans of soda!**

**In this dilemma, you probably added just the milk together and just the soda can together. You know that the milk cartons are alike. You know that soda cans are alike. But the milk and the soda cans are not alike. In mathematics, we are able to combine** *like terms***but we do not combine** *unlike terms***.**

As we already saw, **a term can 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 \begin{align*}4x\end{align*} is 4.** The coefficient of \begin{align*}-7a^2\end{align*} is -7. The coefficient of \begin{align*}y\end{align*} is 1 because its numerical factor is an unwritten. You could write “\begin{align*}1y\end{align*}” to show that the coefficient of \begin{align*}y\end{align*} is 1 but it is not necessary because any number multiplied by 1 is itself.

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

Take a look at some of these.

\begin{align*}7n\end{align*} and \begin{align*}5n\end{align*} are like terms because they both have the variable \begin{align*}n\end{align*} with an exponent of 1.

\begin{align*}4n^2\end{align*} and \begin{align*}-3n\end{align*} are not like terms because, although they both have the variable \begin{align*}n\end{align*}, they do not have the same exponent

\begin{align*}5x^3\end{align*} and \begin{align*}8y^3\end{align*} are not like terms because, although they both have the same exponent, they do not have the same variable.

**Like terms can be combined by adding their coefficients.**

\begin{align*}7n+5n &=12n\\ 3x^3+5x^3 &=8x^3\\ -2t^4-10t^4 &=-12t^4\\ 2n^2-3n+5n^2+11n &=7n^2+8n\end{align*}

Notice that the exponent does not change when you combine like terms. If you think of \begin{align*}7n\end{align*} as simply a shorter way of writing \begin{align*}n + n + n + n + n + n + n\end{align*} and \begin{align*}5n\end{align*} as a shorter way of writing \begin{align*}n + n + n + n + n\end{align*}, then combining those like terms to get \begin{align*}12n\end{align*} is a simpler way to write \begin{align*}7n + 5n\end{align*}.

Combine like terms.

#### Example A

\begin{align*}2x - 8y - 4x + 7y + 9\end{align*}

**Solution: \begin{align*}-2x-y+9\end{align*}**

#### Example B

\begin{align*}5a+3b-8b+a-7\end{align*}

**Solution: \begin{align*}6a-5b-7\end{align*}**

#### Example C

\begin{align*}5a-7b+8b-2a+8a-9+8\end{align*}

**Solution: \begin{align*}11a+b-1\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept.

Here is the problem that Jessie is stuck on.

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

She can combine the \begin{align*}x\end{align*}'s and the \begin{align*}y\end{align*}'s.

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

Now let's put it altogether.

\begin{align*}-4x+4y\end{align*}

**This is our answer.**

### Vocabulary

- Polynomial
- an algebraic expression that shows the sum of monomials. A polynomial can also be named when there are more than three terms present.

- Monomial
- an expression where there is one term.

- Binomial
- an expression where there are two terms.

- Trinomial
- an expression where there are three terms.

- Constant
- a term that is a single number such as 4 or 9.

- Coefficient
- a variable and a numerical factor and the numerical factor is the coefficient

- Like Terms
- are terms that have the same variables and same exponents.

### Guided Practice

Here is one for you to try on your own.

Simplify by combining like terms.

\begin{align*}15x-12x+3y-8x+7y-1+5\end{align*}

**Solution**

First, let's combine the like terms.

\begin{align*}15x-12x-8x &= -5x \\ 3y+7y &= 10y \\ -1+5 &= 4\end{align*}

Now let's put it altogether.

\begin{align*}-5x+10y+4\end{align*}

**This is our answer.**

### Video Review

### Practice

Directions: Simplify the following polynomials by combining like terms.

- \begin{align*}6x+7-18x+4\end{align*}
- \begin{align*}5x-7x+5x+4-9\end{align*}
- \begin{align*}3x+8y-5x+3y\end{align*}
- \begin{align*}17x^2-7x^2-5x+3x+14\end{align*}
- \begin{align*}3xy-9xy-5x+4x-7+3\end{align*}
- \begin{align*}9x+7y-15x+4x-9y\end{align*}
- \begin{align*}3x+7-5x-8y+4x-2y+7\end{align*}
- \begin{align*}3xy-xy-15x+4-11\end{align*}
- \begin{align*}-8x+3x+7y-5x+4y-2\end{align*}
- \begin{align*}3x^2+6x-3y+2x-7\end{align*}
- \begin{align*}14xy-18xy+7y+8x-2x+9\end{align*}
- \begin{align*}3x+7-5x+4y-18y\end{align*}
- \begin{align*}6y^2-4y^3+y^2-8\end{align*}
- \begin{align*}-5q+q^2+7-q-7\end{align*}
- \begin{align*}n^2m-3n^2m+5n^2 m^2+11n\end{align*}

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.Coefficient

A coefficient is the number in front of a variable.constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as*x, y*or

*a*.

like terms

Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.Monomial

A monomial is an expression made up of only one term.Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Trinomial

A trinomial is a mathematical expression with three terms.### Image Attributions

Here you'll simplify polynomials by combining like terms.

## Concept Nodes:

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.Coefficient

A coefficient is the number in front of a variable.constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as*x, y*or

*a*.

like terms

Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.Monomial

A monomial is an expression made up of only one term.Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Trinomial

A trinomial is a mathematical expression with three terms.