Polynomials: Combining "Like" Terms

The terms which can be combined are called "like" terms; terms which cannot be combined are called "unlike" terms.

Which terms can be combined, and why?

Terms can be combined only if they have the exact same variable portion. And, by "exact same", I mean "the same variable(s), raised to the same power(s)". Here is a rundown of what's what:

So, to decide if two terms are "like" terms that can be combined, we look at the variable portion of those terms. The numerical portion of two terms is what will get combined (as we'll see shortly), but it's the variable portion of those two terms that determines if the terms are combine-able. To be able to be combined, the terms' variable portions must contain the exact same variable(s) with the exact same power(s).

Once you have determined that two terms are indeed "like" terms and can indeed therefore be combined, you can then deal with the terms in a manner similar to what you did in grammar school. When you were first learning to add, you would do "five apples and six apples is eleven apples". You have since learned that, as they say, "you can't add apples and oranges". That is, "five apples and six oranges" is just a big pile of fruit; it isn't something like "eleven applanges".

Combining like terms works much the same way; we add the numerical parts, while carrying along the variable parts, almost like a unit, or like the "apples" we'd added.

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• Simplify 3x + 4x

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• Simplify 2x² + 3x − 4 − x² + x + 9

In the second line above, many students find it helpful to write in the understood coefficient of 1 in front of any variable expressions having no written coefficient, as is shown in red below:

It is not required that the understood 1 be written in when simplifying expressions like this, but many students find this technique to be very helpful, at least when they're starting out. You should use whatever method helps you to consistently complete the simplification correctly.

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• Simplify 10x³ − 14x² + 3x − 4x³ + 4x − 6

Warning: When moving the terms around, remember that the terms' signs move with them. Don't mess yourself up by leaving orphaned "plus" and "minus" signs behind.

If it helps you keep things straight, rewrite the expression like this:

By turning the subtractions into additions of negatives, it is clear where the "minus" signs belong, and it's easier to move them correctly:

...and so forth. Do what works for you.

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• Simplify 25 − (x + 3 − x²)

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I'm going to stress here again the fact that very many students, at least initially, have trouble with the "minus" signs in front of parentheticals. If it helps you to keep track of what's going on, try putting the "understood" 1 in front of the parentheses (highlighted in red below), and then multiplying that constant through onto the terms within the parenthetical, as shown below:

While the first format (without the understood 1's being written in) is the more "standard" format, either format is mathematically valid. You should use the format that works most successfully for you.

Don't be shy about inserting the understood 1 when you're starting out; don't feel bound to continue using it once you get to feeling confident without it. You do you.

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• Simplify x + 2(x − [3x − 8] + 3)

Just so you know, this is the kind of problem that us math teachers love to put on tests (yes, some of us are kinda sick puppies), so you should expect to need to be able to deal with nested grouping symbols like this.

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• Simplify [(6x − 8) − 2x] − [(12x − 7) − (4x − 5)]

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• Simplify −4y − [3x + (3y − 2x + {2y − 7}) − 4x + 5]

The simplification above involved two different variables. It is customary (though not technically required) to list the terms so that their variables are in alphabetical order. That's why I listed the 3x before the 9y. And, regardless of what might be going on with the variable terms, the constant term always goes at the very end.

(If you think you need more practice with this last type of problem (with all the brackets and the negatives and the parentheses, then you might want to review the "Simplifying with Parentheses" lesson.)

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When we're talking apples and oranges, it's usually pretty easy to see what should be combined with what else, and in what manner. But once we start dealing with variables, things can get confusing. Don't get careless and confuse multiplication and addition. This may sound like a silly thing to me to say, but it may be the most commonly-made student mistake when combining like terms (aside from messing up the order of operations).

So if you have something like x³ + x², do not try to say that this somehow equals something like x⁵ or 5x. The terms are unlike, and cannot be combined. If you have something like 2x + x, do not try to say that this somehow equals something like 2x². These terms are like terms, and are combined by adding their coefficients, to get 3x.

Take your time, and make sure you are keeping straight in your head how multiplication works, versus how addition works. In fact, "combining like terms" is a topic for which it would be difficult to do "too much" practice. Do as many practice problems as you can. You want proper simplification to become so familiar that the process of it almost flows, so you're barely even having to think about what you're doing. This will stand you in good stead for the rest of your mathematical (and science-based) career! 👍