Combining like terms. This subject seems really complicated, but it’s really not. It just follows a couple of rules. The first rule is the one most of you probably already know, PEMDAS.

Parentheses (x + y)

Exponents x^{2}

Multiplication x*y

Division x/y

Addition x + y

Subtraction x – y

A like term is any term that can be grouped together. For example:

2x + x + 5x + 9y + 2y

In the expression above, each term with x and each term with y is a like term. Grouping the like terms gets thus:

8x + 11y

Remember, multiplication is just repeated addition. So another way of thinking of the above process is like this:

2x + x + 5x + 9y + 2y

(x + x) + x + (x + x + x + x + x) + (y + y + y + y + y + y + y + y + y) + (y + y)

x + x + x + x + x + x + x + x + y + y + y + y + y + y + y + y + y + y + y

(x + x + x + x + x + x + x + x) + (y + y + y + y + y + y + y + y + y + y + y)

8x + 11y

Now, another important part of combining like terms is exponents. Just like how multiplication is repeated addition, exponents are repeated multiplication, and can be combined:

x^{2}*x*x^{5}*y^{9}*y^{2}

= x*x*x*x*x*x*x*x*y*y*y*y*y*y*y*y*y*y*y

= x^{8}*y^{11}^{}

Repeated occurrences of the same exponent can also be combined:

2x^{2} + 5x^{2}

= x*x + x*x + x*x + x*x + x*x + x*x + x*x

= 7x^{2}

**Important: The above process can only work if the exponents are the same. If does not apply to different exponents.**

x^{3}+9x^{2} **¹** 10x^{5}

Why? It’s really easy to see when expanded:

x^{3}+9x^{2}

= (x*x*x) + (x*x) + (x*x) + (x*x) + (x*x) + (x*x) + (x*x) + (x*x) + (x*x) + (x*x)

Since PEMDAS says that multiplication comes first, all of the x terms have to be multiplied before they can be added, which makes different degrees incompatible.

We can also prove it algebraically by testing a value:

x = 2

x^{3}+9x^{2} = 2^{3}+9*2^{2} = 8 + 9*4 = 8 + 36 = 44

10x^{5} = 10*2^{5} = 10*32 = 320

320 ¹ 44

Now that we have that cleared, let’s look at more complex scenarios.

2x + 9y^{9} + 18 + 95z^{4} + 17y^{5} + 6x + 7x^{2} + 1 + 10y^{9} – 15z^{4} + 8x^{3}

This looks really tricky and complex. But it’s really simple when organized properly.

First, identify all of the terms with the same variable.

2x + 9y^{9} + 18 + 95z^{4} + 17y^{5} + 6x + 7x^{2} + 1 + 10y^{9} – 15z^{4} + 8x^{3}

Then, group those together. Pay attention to what I did with the 15z^{4}; the negative sign has to be moved as well. We previously didn’t cover that since all the terms so far have been positive.

2x + 6x + 7x^{2} + 8x^{3} + 9y^{9} + 17y^{5} + 10y^{9} + 95z^{4} – 15z^{4} + 18 + 1

Then, within each group of variables, sort them into descending exponent order.

8x^{3} + 7x^{2} + 6x + 2x + 10y^{9} + 9y^{9} + 17y^{5} + 95z^{4} – 15z^{4} + 18 + 1

Finally, group into like terms and add.

8x^{3} + 7x^{2} + (6x + 2x) + (10y^{9} + 9y^{9}) + 17y^{5} + (95z^{4} – 15z^{4}) + (18 + 1)

8x^{3} + 7x^{2} + 8x + 19y^{9} + 17y^{5}+ 80z^{4} + 19

Even though like terms can be really tricky, there’s nothing to be afraid of. Just group each variable together, take note of negatives, parentheses, and exponents, and they are actually quite simple. Good luck!