Explainer: Simplifying Expressions: Combining Like Terms

In this explainer, we will learn how to simplify polynomial expressions by adding and subtracting like terms.

Key Information

  1. The Associative Property of Addition
    The associative property of addition tells us that (𝑎+𝑏)+𝑐=𝑎+(𝑏+𝑐).
    In words, this tells us that the order in which you sum the numbers or terms in an expression does not change the value of the expression.
  2. The Commutative Property of Addition
    The commutative property of addition tells us that 𝑎+𝑏=𝑏+𝑎.
    In words, this tells us that changing the order of terms in an expression does not change the value of the expression.
  3. The Distributive Property
    The distributive property tells us that 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐.
    In words, this tells us that we can distribute 𝑎 throughout the parentheses by multiplying each of the terms 𝑏 and 𝑐 by 𝑎 separately and finding the sum of the result.

When answering questions involving the simplification of polynomial expressions, we often use the properties stated above. We will work through a couple of examples where we formally state the algebraic properties that we are using, as we may face questions where we are asked specifically about these properties. Equally, we will work through some examples where we use the different properties but do not necessarily state them, as, in practice, this is how you develop fluency in your use of algebra.

Let us start with two formal examples.

Example 1: Simplifying Polynomial Expressions

Simplify 65𝑧+22(𝑧4).

Answer

We start by using the distributive property to expand each of the sets of parentheses. We need to be particularly careful with the signs of the resulting multiplications. Dealing with each of the two sets of parentheses separately, we have that 65𝑧+2=(6)5𝑧+(6)(2)=30𝑧+12,2(𝑧4)=(2)(𝑧)+(2)(4)=2𝑧+8.

If we sum the two resulting expressions, we get 30𝑧+122𝑧+8.

Now, we can use the commutative property to reorder the terms of the expression and the associative property to group like terms: 30𝑧2𝑧+(12+8).

At this point, we can simplify any like terms to get 30𝑧2𝑧+20.

Example 2: Simplifying Polynomial Expressions

Simplify 6𝑥𝑥+𝑦𝑦6𝑥𝑦+6𝑦𝑥.

Answer

We start by using the distributive property to expand each of the three sets of parentheses. We need to be particularly careful with the signs of the resulting multiplications. Dealing with each of the three sets of parentheses separately, we have that 6𝑥𝑥+𝑦=(6𝑥)𝑥+(6𝑥)(𝑦)=6𝑥+6𝑥𝑦,𝑦6𝑥𝑦=(𝑦)(6𝑥)+(𝑦)𝑦=6𝑥𝑦+𝑦,6𝑦𝑥=6𝑦+6𝑥=6𝑦6𝑥.

If we sum the three resulting expressions, we get 6𝑥+6𝑥𝑦6𝑥𝑦+𝑦+6𝑦6𝑥.

Now, we can use the commutative property to reorder the terms of the expression and the associative property to group like terms: 6𝑥6𝑥+(6𝑥𝑦6𝑥𝑦)+𝑦+6𝑦.

At this point, we can simplify each of the pairs of like terms to get 7𝑦.

Now, let us look at some examples that we will answer by taking a less formal approach.

Example 3: Simplifying Polynomial Expressions

Simplify 6(3𝑏+2)+4(2𝑏+4).

Answer

Our first step in solving this problem is to expand the parentheses: 18𝑏+12+8𝑏+16.

If we now collect all the like terms, our expression simplifies to our answer which is 26𝑏+28.

Example 4: Simplifying Polynomial Expressions

Simplify 3𝑥[𝑥+(𝑦𝑥)]+𝑦[2𝑦+2(𝑥𝑦)], and find the value of the expression when 𝑥=𝑦=1.

Answer

Our first step in solving this problem is to remove the parentheses inside the brackets. For the first set of parentheses, the expression does not change as we are distributing positive one throughout the brackets. The second set of parentheses, however, needs to be expanded as we need to distribute two throughout the bracket: 3𝑥[𝑥+𝑦𝑥]+𝑦[2𝑦+2𝑥2𝑦].

If we now collect all the like terms within the brackets, our expression simplifies to 3𝑥[𝑦]+𝑦[2𝑥].

We then multiply out the brackets to get 3𝑥𝑦+2𝑥𝑦, which simplifies to 5𝑥𝑦.

To finish, we substitute 𝑥=𝑦=1 which give us 5(1)(1)=5.

Example 5: Simplifying Polynomial Expressions

Expand the brackets and simplify 7𝑥(𝑥+𝑦)+𝑦[3𝑥4𝑦(𝑥+2𝑦)]+7𝑥[3𝑥2(𝑥+3𝑦)].

Answer

If we start by expanding all the parentheses by distributing each of the terms with which they are multiplied, we get 7𝑥+7𝑥𝑦+𝑦3𝑥4𝑥𝑦8𝑦+7𝑥[3𝑥2𝑥6𝑦].

If we now collect the like terms within the brackets, our expression simplifies to 7𝑥+7𝑥𝑦+𝑦3𝑥4𝑥𝑦8𝑦+7𝑥[𝑥6𝑦].

We then expand the two brackets by distributing the 𝑦 and 7𝑥 throughout their respective brackets to get 7𝑥+7𝑥𝑦+3𝑥𝑦4𝑥𝑦8𝑦+7𝑥42𝑥𝑦.

We can then simplify by collecting any like terms to get 14𝑥4𝑥𝑦32𝑥𝑦8𝑦.

An example of how you could show your working is shown here:

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