In this explainer, we will learn how to simplify polynomial expressions by adding and subtracting like terms.
Algebraic expressions are an incredibly useful tool for representing relationships between one or more unknowns. These types of relationships appear all throughout mathematics and in many other disciplines as well.
There are many ways of manipulating these expressions to simplify the relationships and even solve for the unknown values. For example, if we are told that doubling a number and adding 3 gives 11, we can call the unknown number and say that .
We can then solve this equation for by subtracting 3 from both sides to get
Then, we divide both sides of the equation through by 2 to get
We apply the same operations to both sides of the equation to isolate the unknown .
More formally, we are actually using properties of numbers to rewrite this relationship. When we subtract 3 from both sides of the equation, we are using the additive inverse property, and when we divide both sides of the equation by 2, we are using the multiplicative inverse property.
We can use these properties since the unknown value is a number (called a variable). So, we can treat is like a number. We can extend this idea to find more ways of rewriting expressions involving unknowns.
To see an example of this, letβs first consider an example in which we have a field containing cows and sheep and we want to write a mathematical expression for the total number of animals. If we count 4 cows and 5 sheep in the field, then, by using the variable for cows and for sheep, we will write the expression to denote the total number of animals in the field.
We note that this expression cannot be simplified any further since and denote different things. This is because they are not like terms (i.e., they are unlike terms).
On the other hand, what if 3 cows were added to the field? Mathematically, we would write the new total number as .
However, this expression is not in its simplest form. By changing the order, we add the numbers together (i.e., by using the commutative property of addition). We can rewrite this as . We can note that 4 cows added 3 cows will be the same as 7 cows.
We call the terms and like terms since they are both amounts of the same variable, .
Like terms can be added together by combining the coefficients as follows:
Thus, the simplified form of the total number is .
We can define like terms formally as follows.
Definition: Like Terms
We say that two terms are like terms if they both have the same variables raised to the same powers.
In our first example, we will use this idea of combining like terms by adding or subtracting their coefficients to simplify a given algebraic expression.
Example 1: Combining Like Terms to Simplify an Algebraic Expression in One Variable
Simplify the expression .
Answer
We are asked to simplify an algebraic expression with four terms. We first notice that every term has a single unknown factor of . Since every term has the same variables raised to the same the powers, we can say that all four terms are like terms.
We then recall that we can add or subtract any number of like terms by adding or subtracting their coefficients where the variables and their powers remain unchanged. We can do this by taking out the shared factor of as follows:
Now, we evaluate the expression in the coefficient to get
In the above example, we started with an expression with four terms and simplified it into one with only a single term. In general, we can use this process to reduce the number of terms in an algebraic expression with like terms. For example, if we had a polynomial expression with like terms, then we can use this method to reduce the number of terms in the polynomial expression.
However, we cannot always reduce the number of terms using like terms. For example, consider the expression . Since the powers of are different in the two terms, they are not like terms. In this case, we cannot combine the two terms, and they must remain separate.
We can state the property that allows us to add and subtract like terms as follows.
Property: Adding or Subtracting Like Terms
We can add or subtract any number of like terms by adding or subtracting their coefficients. The variables and their powers remain unchanged.
We can prove this property more formally by using the distributive property of multiplication over addition, which states that for any rational numbers , , and we have
If is the shared factor of the like terms, then combining like terms is a direct application of this property.
For example, letβs say that , , and . Then, the distributive property tells us that
Letβs now see an example of finding a polynomial expression and then reducing the number of terms in the expression by combining like terms.
Example 2: Solving a Geometric Problem by Combining Like Terms in an Algebraic Expression
If the area of each shape is given by the expression written inside it, write a simplified expression for the sum of their areas.
Answer
We first need to add all of the expressions for the four given shapes together. This gives us
We now want to simplify this expression. We start by checking for like terms. We recall that these are terms that have the same variables raised to the same the powers. We see that only the second and third terms are like terms since the first does not have a factor of and the final term has a factor of .
We then recall that we can combine like terms by combining their coefficients. This gives us
Finally, we can use the commutativity property of addition to rewrite this expression in descending powers of . This gives us
Thus far, we have only dealt with examples in a single variable. However, we can combine multiple like terms to simplify an expression as we will see in our next example.
Example 3: Combining Like Terms to Simplify an Algebraic Expression in Two Variables
Simplify the expression .
Answer
We can simplify algebraic expressions by combining like terms, which we recall are terms that have the same variables raised to the same the powers. We see that and only have a single variable and and only have a single variable . Thus, these are each a pair of like terms.
We can combine like terms by combining their coefficients. We can rearrange the terms using commutativity to get
Taking out the shared factors of and , we have
In our next example, we will simplify an expression for an amount of water in terms of full bottles and full glasses.
Example 4: Solving a Real-World Problem by Combining Like Terms in an Algebraic Expression
In the following diagram, is the quantity of water in a full bottle and is that in a full glass. The total quantity of water shown in the diagram can be expressed as . Simplify this expression.
Answer
We can simplify algebraic expressions by combining like terms, which we recall are terms that have the same variables raised to the same the powers. In this case, the -terms are like terms and the -terms are like terms since they have the same variables raised to the same exponents so that the like terms are together.
We add like terms by adding their coefficients. We can start by using the commutativity of addition to reorder the terms:
Next, by noting that and , we take out the shared factors:
Now, we want to evaluate the coefficients. To add fractions, we need to rewrite them with the same denominator. We have
Since the coefficients represent the amount of water in terms of bottles and glasses, respectively, it may be a good idea to write the coefficients as mixed numbers. We have
Hence,
In our next example, we will simplify an algebraic expression involving three different variables.
Example 5: Combining Like Terms to Simplifying an Algebraic Expression in Three Variables
Simplify .
Answer
We can simplify algebraic expressions by combining like terms, which we recall are terms that have the same variables raised to the same the powers. We first need to identify the like terms.
We see that and are like terms since they share a single nonnumeric factor of only . Similarly, and are like terms since they share a single nonnumeric factor of only . Finally, and are like terms since they share a single nonnumeric factor of only .
We can reorder the terms using the commutativity of addition to get
Now, we can combine the like terms by taking out the shared algebraic factors. We get
Finally, we evaluate the coefficients as follows:
In our final example, we will construct an algebraic expression from a given geometric problem and then simplify the expression by combining like terms.
Example 6: Solving a 3D Geometric Problem by Combining Like Terms in an Algebraic Expression
Express, in terms of , , and , the sum of the surface areas of the two given figures.
Answer
We first need to find expressions for the surface area of each shape. Since they are cuboids, each face is a rectangle. So, the area of each face is the product of its sides.
Opposite faces have the same area, so we can double the area of each face to find the sum of the areas of the opposite faces. Thus, if we double and add the expressions, we will find the surface area.
The first shape has area
The second shape has area
We now want to add these expressions together. We get
We can now simplify this expression by combining like terms, which we recall are terms that have the same variables raised to the same powers. We can rearrange the expression so that the pairs of like terms are together (using the commutative property) to get
We can then take out the shared factors to get
Finally, we evaluate the sums inside of the parentheses to get
Key Points
- We say that two terms are like terms if they both have the same variables raised to the same the powers.
- We can add or subtract any number of like terms by adding or subtracting their coefficients. The variables and their powers remain unchanged.
- We can combine like terms in a polynomial expression to reduce the number of terms in the expression.