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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
License: CC BY-NC 3.0

Jessie is stuck on her math homework. She is stuck on the problem \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: \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 \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 number 1. 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.

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

\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*}\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 \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*}.

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:

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

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

Example 2

Simplify by combining like terms:

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

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

\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:

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

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

Example 3

Simplify by combining like terms.

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

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

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

Then rewrite the terms in a single expression:

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

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

Example 4

Simplify by combining like terms.

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

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:

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

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

Example 5

Simplify by combining like terms.

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

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:

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

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

Review

Simplify the following polynomials by combining like terms.

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3. 

Vocabulary

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

  1. [1]^ License: CC BY-NC 3.0

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