Simplifying with Parentheses: An Introduction

What is a parenthetical expression?

A parenthetical expression is a mathematical expression, usually having two or more terms, which is contained within a pair of parentheses and is itself contained within a larger expression. Typically, you need to multiply some value through those parentheses in order to be able to remove the grouping symbols; that is, you'll apply the Distributive Property.

• 3 + x is not a parenthetical expression, because there are no parentheses.
• (3 + x) contains the mathematically trivial parenthetical expression (3 + x). ("Mathematically trivial" here means "it technically fulfills the definition, but is otherwise a boring special case of that definition. Yes, there is an expression contained within a pair of parentheses, but the grouping symbols are entirely unnecessary.)
• 2(3) contains the mathematically trivial parenthetical expression "(3)". The multiplication could also be stated as 2 × 3. (Multiplication indicated by a value tacked onto a parenthetical, rather than with a "times" sign, is called "multiplication by juxtaposition".)
• 2(3 + x) contains the parenthetical expression (3 + x).
• −(3 + x) contains the parenthetical expression (3 + x). (The "minus" in front of the parenthetical may also be expressed as −1: −1(3 + x).)

What is "simplifying with parentheses"?

"Simplifying with parentheses" refers to the process simplifying an expression containing parenthetical expressions (that is, containing mathematical expressions which are enclosed within grouping symbols). The simplification process uses the Distributive Property to multiply exterior values through the parentheses, thus allowing the grouping symbols to be removed. You will be "distributing [something] over" (that is, multiplying [something] through) something else that is inside a set of parentheses.

I will walk you through some examples of increasing difficulty, and you should note, as this lesson progresses, the importance of "simplifying as you go" and of doing each step neatly, completely, and exactly.

• Simplify 3(x + 4)

Written all in one line, the above would look like the below:

(I would not recommend trying to work sideways like the above. Working line-by-line usually has a greater chance of success.)

The error most commonly made at this stage is to take the 3 through the parentheses, but only onto the x, forgetting to carry it through onto the 4 as well. If you need to draw little arrows to help you remember to carry the multiplier through onto everything inside the parentheses, then draw them!

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• Simplify −2(x − 4)

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A common mistake students with this type of problem is to lose a "minus" sign somewhere. For example, somebody might do the following:

Did you notice how the "−4" originally inside the parenthetical somehow turned into a plain old "4" when the −2 went through the parentheses? That's why the answer in the gray box above ended up being wrong. (Compare with the correct answer, −2x + 8.) Be careful with the "minus" signs! Until you are confident in your skills, take the time to write out the distribution, complete with the signs, as I did.

Another way to keep track of what's going on is to write out the multiplications explicitly. Below, I've used color, where the "−2" is in red, the "− 4" is in blue, the variable "x" is left as black text, and the "8" in the answer is in purple:

If you have difficulty with the subtraction, then try converting it into the addition of a negative:

Do as many steps as you need to do, in order to consistently get the correct answer. Whatever works for you is "the right way" to do these!

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• Simplify −(x − 3)

Take note that the expressions "−1x + 3" and "−x + 3" are technically the same thing; in my classes, either would be perfectly acceptable as your hand-in answer.

However, some teachers will accept only "−x + 3" and would count "−1x + 3" as not fully simplified. (In other words, they will insist that the "understood" 1s be dropped.)

It would be wise to check with your instructor, especially if you find it helpful to write in that understood "1".

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• Simplify 2 + 4(x − 1)

If I were your instructor, I would accept either of "4x − 2" and "−2 + 4x" as a valid answer. However, most texts expect the answer to be written with its terms in descending order (which, in this case, means with the variable term first, followed by the plain number).

You should know that the two expressions of the answer are mathematically exactly the same, while keeping in mind that some instructors insist that the answer be written in descending order for that answer to be considered to be completely "correct". It would probably be best to get in the habit now of writing your answers in descending order.