12.3: Simplify Polynomials by Combining Like Terms
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 3x8+4x^5 \qquad 7a^2+9b4b^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—“fiveterm 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*}
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*}
\begin{align*}4n^2\end{align*}
\begin{align*}5x^3\end{align*}
Like terms can be combined by adding their coefficients.
\begin{align*}7n+5n &=12n\\ 3x^3+5x^3 &=8x^3\\ 2t^410t^4 &=12t^4\\ 2n^23n+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*}
Combine like terms.
Example A
\begin{align*}2x  8y  4x + 7y + 9\end{align*}
Solution: \begin{align*}2xy+9\end{align*}
Example B
\begin{align*}5a+3b8b+a7\end{align*}
Solution: \begin{align*}6a5b7\end{align*}
Example C
\begin{align*}5a7b+8b2a+8a9+8\end{align*}
Solution: \begin{align*}11a+b1\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*}
\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*}15x12x+3y8x+7y1+5\end{align*}
Solution
First, let's combine the like terms.
\begin{align*}15x12x8x &= 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+718x+4\end{align*}
6x+7−18x+4 
\begin{align*}5x7x+5x+49\end{align*}
5x−7x+5x+4−9 
\begin{align*}3x+8y5x+3y\end{align*}
3x+8y−5x+3y  \begin{align*}17x^27x^25x+3x+14\end{align*}
 \begin{align*}3xy9xy5x+4x7+3\end{align*}
 \begin{align*}9x+7y15x+4x9y\end{align*}
 \begin{align*}3x+75x8y+4x2y+7\end{align*}
 \begin{align*}3xyxy15x+411\end{align*}
 \begin{align*}8x+3x+7y5x+4y2\end{align*}
 \begin{align*}3x^2+6x3y+2x7\end{align*}
 \begin{align*}14xy18xy+7y+8x2x+9\end{align*}
 \begin{align*}3x+75x+4y18y\end{align*}
 \begin{align*}6y^24y^3+y^28\end{align*}
 \begin{align*}5q+q^2+7q7\end{align*}
 \begin{align*}n^2m3n^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
Description
Learning Objectives
Here you'll simplify polynomials by combining like terms.