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# Simplify Expressions

## Simplifying expressions refers to the concept of simplifying mathematical expressions that have exponents. Learn more from our resources below.

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Simplify Polynomials by Combining Like Terms

Jessie is stuck on her math homework. She is stuck on the problem 5x3y9x+7y\begin{align*}5x-3y-9x+7y\end{align*}. 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: x2+53x8+4x57a2+9b4b3+6\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 4x\begin{align*}4x\end{align*} is 4. The coefficient of 7a2\begin{align*}-7a^2\end{align*} is -7. The coefficient of y\begin{align*}y\end{align*} is 1 because its numerical factor is an unwritten number 1. You could write “1y\begin{align*}1y\end{align*}” to show that the coefficient of y\begin{align*}y\end{align*} is 1 but it is not necessary because any number multiplied by 1 is itself.

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

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

4n2\begin{align*}4n^2\end{align*} and 3n\begin{align*}-3n\end{align*} are not like terms because, although they both have the variable n\begin{align*}n\end{align*}, they do not have the same exponent.

5x3\begin{align*}5x^3\end{align*} and 8y3\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.

7n+5n3x3+5x32t410t42n23n+5n2+11n====12n8x312t47n2+8n\begin{align*}\begin{array}{rcl} 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{array}\end{align*}

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

### Examples

#### Example 1

Here is the problem that Jessie is stuck on:

5x3y9x+7y\begin{align*}5x - 3y - 9x + 7y\end{align*}

First, look at the like terms and combine them:

5x9x3y+7y==4x4y\begin{align*}\begin{array}{rcl} 5x - 9x &=& -4x \\ -3y + 7y &=& 4y \end{array}\end{align*}

Then write the terms in a single expression again:

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

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

#### Example 2

Simplify by combining like terms:

15x12x+3y8x+7y1+5\begin{align*}15x-12x+3y-8x+7y-1+5\end{align*}

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

15x12x8x3y+7y1+5===5x10y4\begin{align*}\begin{array}{rcl} 15x - 12x - 8x &=& -5x \\ 3y + 7y &=& 10y \\ -1 + 5 &=& 4 \end{array}\end{align*}

Then rewrite the combined terms in a single expression:

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

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

#### Example 3

Simplify by combining like terms.

2x8y4x+7y+9\begin{align*}2x- 8y - 4x + 7y + 9\end{align*}

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

2x4x8y+7y9==2xy\begin{align*}\begin{array}{rcl} 2x-4x &=& -2x \\ -8y + 7y &=& -y \\ 9 \end{array}\end{align*}

Then rewrite the terms in a single expression:

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

The answer is 2xy+9\begin{align*}-2x - y + 9\end{align*}.

#### Example 4

Simplify by combining like terms.

5a+3b8b+a7\begin{align*}5a + 3b - 8b + a -7\end{align*}

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

5a+a3b8b7==6a5b\begin{align*}\begin{array}{rcl} 5a + a &=& 6a \\ 3b - 8b &=& -5b \\ -7 \end{array}\end{align*}

Then rewrite the terms in a single expression:

6a5b7\begin{align*}6a-5b-7\end{align*}

The answer is 6a5b7\begin{align*}6a-5b-7\end{align*}.

#### Example 5

Simplify by combining like terms.

5a7b=8b2a+8a9+8\begin{align*}5a - 7b = 8b - 2a + 8a - 9 + 8\end{align*}

First, look at the like terms and combine them.

5a2a+8a7b+8b9+8===11ab1\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:

11a+b1\begin{align*}11a + b - 1\end{align*}

The answer is 11a+b1\begin{align*}11a + b - 1\end{align*}.

### Review

Simplify the following polynomials by combining like terms.

1. 6x+718x+4\begin{align*}6x + 7 - 18x + 4\end{align*}
2. 5x7x+5x+49\begin{align*}5x - 7x + 5x + 4 -9\end{align*}
3. 3x+8y5x+3y\begin{align*}3x + 8y - 5x + 3y\end{align*}
4. 17x27x25x+3x+14\begin{align*}17x^2 - 7x^2 - 5x + 3x + 14\end{align*}
5. 3xy9xy5x+4x7+3\begin{align*}3xy - 9xy - 5x + 4x - 7 + 3\end{align*}
6. 9x+7y15x+4x9y\begin{align*}9x + 7y - 15x + 4x - 9y\end{align*}
7. 3x+75x8y+4x2y+7\begin{align*}3x + 7 - 5x - 8y + 4x - 2y + 7\end{align*}
8. 3xyxy15x+411\begin{align*}3xy - xy - 15x + 4 -11\end{align*}
9. \begin{align*}-8x + 3x + 7y - 5x + 4y - 2\end{align*}
10. \begin{align*}3x^2 + 6x - 3y + 2x - 7\end{align*}
11. \begin{align*}14xy - 18xy + 7y + 8x - 2x + 9\end{align*}
12. \begin{align*}3x + 7 - 5x + 4y - 18y\end{align*}
13. \begin{align*}6y^2 - 4y^3 + y^2 - 8\end{align*}
14. \begin{align*}-5q + q^2 + 7 - q -7\end{align*}
15. \begin{align*}n^2m - 3n^2m + 5n^2m^2 + 11n\end{align*}

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### Vocabulary Language: English

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.