Getting Rid of Parentheses—Who Needs 'Em?
We've seen a lot of expressions with parentheses in them on this algebraic journey of ours. We prefer them as smiley faces in emoticons—the rest of the time, they sort of confuse things. Here's a valid, but horrid, expression:
((x + 1)((3 – 4x)(4x + 2) – 5))2
While this expression makes mathematical sense, the high number of parentheses in it makes it hard to tell what's going on. In terms of English, how would you like to read this sentence?
"This sentence (the one you are reading (as if we would be referring to any other sentence) right now (as opposed to later)) is so (poorly) constructed (put together (like a building with an unstable foundation)), it makes me (literally) want to vomit."
Not pretty, right? To make things clearer, we can use the symbols { } or [ ] in place of some of the parentheses ( ). While still not the most pleasant-looking expression we've ever seen, at least it's a little easier to figure out what's being multiplied by what in this rewritten expression:
[(x + 1){(3 – 4x)(4x + 2) – 5}]2
It is okay to think of ( ), { }, and [ ] all as parentheses, since they're all used for the same purpose. Because these symbols serve to group terms together, we also call them grouping symbols. The person who uses these symbols is called the grouper. He tastes delicious when broiled and seasoned with ground white pepper and paprika.
When we're given an expression with a plethora of parentheses, one way to rewrite the expression is to eliminate parentheses until they're all gone. Unfortunately, there's no special "Parentheses-B-Gone" spray you can buy at CVS. Getting rid of parentheses often involves using the distributive property.
Sample Problem
Eliminate parentheses in the expression 4{x + 2(3 – x2)}.
First way: use the distributive property on the inner set of parentheses to get 4{x + 6 – 2x2}. Then distribute the remaining set of parentheses to get 4x + 24 – 8x2. Parentheses, we hardly knew ye.
Second way: use the distributive property on the outer set of parentheses to arrive at 4x + 8(3 – x2), then on the remaining set of parentheses to get 4x + 24 – 8x2. Reassuringly, we got the same answer as we did when we tried it the first way.
Third way: yell "Fire!" and watch them scatter.
As long as we're careful with the arithmetic, it's okay to get rid of parentheses in any order. In fact, it provides a good way to check our work. No matter the order in which we eliminate parentheses, we should get the same answer. We can solve each problem a couple of different ways to make sure we're on the right track. We figured you'd probably want to solve each problem more than once anyway, so this situation works out perfectly.
Be Careful with negative signs, and remember that, to distribute a negative sign over a quantity in parentheses, we erase the negative sign and parentheses while flipping the signs of all the terms in the parentheses. You'll want to flip them about once every thirty seconds to ensure maximum crispiness.