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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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Simplifying Polynomials by Combining Like Terms

Learning Goal

I am learning to represent and simplify a polynomial using a variety of ways.

Warm Up

Do you know how to identify terms that are the same? 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?

Action

Recall that a polynomial is an algebraic expression that shows the sum of monomials.

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

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 pop. In the refrigerator near the cash registers, there are 12 cartons of milk and 26 cans of pop. How many do they have in all?

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

In this dilemma, you probably added just the milk together and just the pop cans together. You know that the milk cartons are alike. You know that pop cans are alike. But the milk and the pop cans are not alike.

Check out this video for another example of combining like terms

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. This tells us that there are a quantity of 4 of the variable x. 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*}.

You may find it helpful to first group the like terms together before simplyfying.

For example

Simplify the following polynomials by collecting like terms and combining them.

a) \begin{align*}2x -4x^2+6+x^2-4+4x\end{align*}

b) \begin{align*}a^3b^3-5ab^4+2a^3b-a^3b^3+3ab^4-a^2b\end{align*}

Solution

a) Rearrange the terms so that like terms are grouped together: \begin{align*}(-4x^2+x^2)+(2x+4x)+(6-4)\end{align*}

Combine each set of like terms: \begin{align*}-3x^2+6x+2\end{align*}

b) Rearrange the terms so that like terms are grouped together: \begin{align*}(a^3b^3-a^3b^3)+(-5ab^4+3ab^4)+2a^3b-a^2b\end{align*}

Combine each set of like terms: \begin{align*}0-2ab^4+2a^3b-a^2b=-2ab^4+2a^3b-a^2b\end{align*}

Now you try ...

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 warm up.

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*}

Consolidation

A polynomial is simplified if it has no terms left that are alike. Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients.

\begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms.

\begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.

If we have a polynomial that has like terms, we simplify by combining them.

\begin{align*}& x^2 + \underline{6xy}-\underline{4xy} + y^2\\ & \qquad \nearrow \qquad \nwarrow\\ & \qquad \text{Like terms}\end{align*}

This polynomial is simplified by combining the like terms \begin{align*}6xy-4xy=2xy\end{align*}. We write the simplified polynomial as \begin{align*}x^2+2xy+y^2\end{align*}.

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*}

Video Review

Here is one for you to try on your own.

How would you identify the following expression?

\begin{align*}3x^2 + 2xy + 3y - 4\end{align*}

This expression has many terms. Therefore, it is called a polynomial (4 - term polynomial).

Word Wall

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.

Practice

Directions: 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*}

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