In order to add or subtract polynomials, you need to know how to recognize like terms. Like terms have exactly the same variables raised to exactly the same exponents. Let’s look at a few examples where we identify like-terms:

Identify the like terms in each set:

a)   3x,   5y,   4x

b)   4x²,   5x,   12x²

c)   2b,  –3a²b,   4,   a²b²,  –10

d)  –2xy,   3x,   11yx

Now let’s try to simplify an expression by combining like terms. To combine like terms (by addition or subtraction), you focus only on the coefficients of those terms. Let’s look at a few examples:

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Combining like terms found in brackets

Sometimes you’ll encounter terms that are placed within parentheses, such as in the example below:

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If the symbol immediately to the left of the bracket is positive (highlighted in blue), you can remove the parentheses completely:

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Notice how both sets of brackets were removed. That’s because in front of (2x + y) (while not shown) is also plus. You’ll have to make this assumption each time. The final answer after combining the terms should be:

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• Some students may find it helpful to first group the like terms before combining them like this: 

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While this may be visually helpful to some, it’s not required.

The next example demonstrates what to do when the symbol in front of the parenthesis is negative. For example:

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The example is nearly identical to the previous one except plus is replaced with minus. If you run into this situation, you can remove the brackets, but the sign of each term inside the brackets must change – a technique known as expanding (this will be covered in greater detail next lesson). Therefore, our expression becomes:

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Notice how 3x was initially positive, it’s now negative; –2y was initially negative, now it’s positive. The brackets around the first two terms were also removed because no symbol was present in front of the bracket. The final answer you should get is:

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Try simplifying the following expressions:

a)   –(a + b) – (a + b)

b)   (3x – 2y + 4z) – (7y + 2w – 5x) – (3w + 2z + 4y – x)