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

### Key Information

**The Associative Property of Addition**

The associative property of addition tells us that .

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.**The Commutative Property of Addition**

The commutative property of addition tells us that .

In words, this tells us that changing the order of terms in an expression does not change the value of the expression.**The Distributive Property**

The distributive property tells us that .

In words, this tells us that we can distribute throughout the parentheses by multiplying each of the terms and by 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 .

### 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

If we sum the two resulting expressions, we get

Now, we can use the commutative property to reorder the terms of the expression and the associative property to group like terms:

At this point, we can simplify any like terms to get

### Example 2: Simplifying Polynomial Expressions

Simplify .

### 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

If we sum the three resulting expressions, we get

Now, we can use the commutative property to reorder the terms of the expression and the associative property to group like terms:

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

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

### Example 3: Simplifying Polynomial Expressions

Simplify .

### Answer

Our first step in solving this problem is to expand the parentheses:

If we now collect all the like terms, our expression simplifies to our answer which is

### Example 4: Simplifying Polynomial Expressions

Simplify , and find the value of the expression when .

### 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:

If we now collect all the like terms within the brackets, our expression simplifies to

We then multiply out the brackets to get which simplifies to .

To finish, we substitute which give us

### Example 5: Simplifying Polynomial Expressions

Expand the brackets and simplify .

### Answer

If we start by expanding all the parentheses by distributing each of the terms with which they are multiplied, we get

If we now collect the like terms within the brackets, our expression simplifies to

We then expand the two brackets by distributing the and throughout their respective brackets to get

We can then simplify by collecting any like terms to get

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