SAT Question: Subtract Gnarly Polynomials

Question

(x²y − 3y² + 5xy²) − (−x²y + 3xy² − 3y²)

Which of the following is equivalent to the expression above?

A) 4x²y²

B) 8xy²−6y²

C) 2x²y+2xy²

D) 2x²y+8xy²−6y²

Answer 1

The SAT will try to scare you with problems that *look* terrifying, but are really not that difficult at all. Remember, there are 25 minutes to solve 20 questions in the no calculator section (where this problem is from). Therefore, a good guideline is to know that every problem has a 1 minute solution (more or less). This one is no exception. Let's walk through my thinking...

(x²y − 3y² + 5xy²) − (−x²y + 3xy² − 3y²)

Ok, subtract a trinomial from another trinomial. We do this by "combining like terms," which means only terms that have the same exact variable and exponent part can be added or subtracted. The others don't mix. I'm just gonna jump right in and grab the first two like terms I see: x²y − −x²y. Remember, two negatives makes a positive so this is equivalent to x²y + x²y, which is 2x²y (1 thing plus 1 thing equals 2 thing). Notice already that answer choices A and B are throw-away answers because they don't contain 2x²y. It's often easy to eliminate 2 answer choices. Now let's proceed. The next terms are −3y² − −3y² (the negatives are gonna trip some people up). This is equivalent to −3y² + 3y², which is 0 (negative thing plus positive thing cancels out). You can see that answer choice D contains a −6y². This is wrong answer bait. You don't even have to finish solving the problem. The answer has to be C. Boom! Bubble it in and move on to the next SAT question. Time is of the essence.

Answer 2

The key to answering this question is to identify like terms. There are 3 sets of like terms: x²y, xy², and y². The next important thing to make sure you do is to distribute the negative to all three terms in the second set of parentheses.

After distributing the negative but before combining like terms we have:

x²y − 3y² + 5xy² + x²y – 3xy² + 3y²

Now combine like terms:

2x²y + 2xy² **Notice that the 3y² cancels out

Therefore, the correct answer choice is: C