Lesson: Simplifying Rational Expressions

Comment on Simplifying Rational Expressions

Hi Brent,

I like the approach of testing values! When can I use this approach in particular please.

Thank you for your advice
Best Regards
gmat-admin's picture

Great question!
Here's an article I wrote that specifically addresses that question: http://www.gmatprepnow.com/articles/data-sufficiency-when-plug-values



[p + 5 + p³(−p − 5)]/(−p − 5) =

A. p + 5 + p³
B. P³ + 5
C. p³
D. p³ - 1
E. p³ - 5

Hi Brent, could you please explain what I am doing wrong in terms of my factoring approach to solve this question?

Step 1: p + 5 + p³(-p - 5)/(-p - 5)
Step 2: [(p + 5) + p³- 1(p + 5)]/(-1)(p + 5)
Step 3: [(p + 5)(1 + p³- 1)]/[(-1)(p + 5)] *In this step I factor out (p + 5) from the numerator
Step 4: (p + 5)(p³)/(-1)(p+5)

Here is where I get stuck.. appreciate your help.
gmat-admin's picture

Question link: https://gmatclub.com/forum/p-5-p-3-p-5-p-95022.html

NOTE: I added some brackets to your solution to reduce ambiguity.

You have a problem going from Step 1 to Step 2.
Notice that (-p - 5) = (-1)(p + 5)
So, Step 2 should look like this: [(p+5) + p³(-1)(p+5)]/(-1)(p+5)
We can simplify this to get: [(p+5) - p³(p+5)]/(-1)(p+5)
From here, we can factor the numerator to get: (p+5)[1 - p³]/(-1)(p+5)
Simplify: [1 - p³]/(-1)
Rewrite as: p³ - 1



Hi Brent - really having a hard time understanding the factoring in the question in this link.

How does the numerator factor out that way? Not sure I see it.

gmat-admin's picture

Question link: https://gmatclub.com/forum/if-xy-0-what-is-the-value-of-x-4-y-2-xy-2-x-3...

Let's start with: x⁴y² - (xy)²
Rewrite as: x⁴y² - (xy)(xy)
Simplify: x⁴y² - x²y²

At this point, we can see that x²y² is the greatest common factor (GCF) of x⁴y² and x²y², which means we can factor it out.
Notice that x⁴y² = (x²y²)(x²)
And x²y² = (x²y²)(1)

So, we get: (x²y²)(x² - 1)

Here's the video on GCF factoring: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...

Does that help?


Yes - I wasn't looking at it from a GCF perspective, but that makes sense. Will just take some practice on getting used to seeing that opportunity to factor out.

One more additional question...why can't I factor out one of the x's in the numerator with the x from the denominator?
gmat-admin's picture

We have: (x² - 1)/x
We can't factor (in a nice way) the x out of x²-1
In order to do so, we must be able to write: x² - 1 = x(something)

We COULD so do like x² - 1 = x(x - 1/x), in which case we get:
(x² - 1)/x = x(x - 1/x)/x = x - 1/x

So, (x² - 1)/x and x - 1/x are equivalent expressions.

Does that help?


Yes, this is helpful. Thank you.

When I got to (x² - 1)/x I ultimately looked at this as the difference of two squares.
For the DS question here ( https://gmatclub.com/forum/if-xy-0-what-is-the-value-of-x-4-y-2-xy-2-x-3-y-283248.html ) the answer A was correct but had it been a PS I probably would have gotten it wrong. I ended up with (x+1)(x-1)/x, which evaluated to 3/2. However, your response to the questions evaluates to 5/2. Could you help me see what I miss, please?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-xy-0-what-is-the-value-of-x-4-y-2-xy-2-x-3...

You did everything perfectly. It turns out that I don't know how to calculate 4-1 (-:

You're absolutely correct to say that the expression evaluates to be 3/2 (not 5/2)

I've edited my solution accordingly.

Thanks for the heads up!


Hi Brent


For the above question I plugged in the value as 1 and I yielded the ans D is there something that I am missing
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-1-2-then-6x-3-3x-2-8x-4-2x-322396.html

When x = 1, the original expression evaluates to -1.
So for each answer choice, we'll replace x with 1 and evaluate each expression to get....
A) -3.5
B) 0.5
C) -1
D) -1
E) 7
Since A, B and E don't evaluate to -1 when x = 1, we can eliminate them.

This means we need to try another value of x.

When x = 2, the original expression evaluates to 8.
Now, for the remaining two answer choices (C and D), we'll replace x with 2 and evaluate each expression to get....
C) 8
D) 2
Since D doesn't evaluate to 8 when x = 2, we can eliminate it.

Answer: D

Does that help?

Hi Brent, yes this helps, but does this mean we will have to test all choices? Or is there a safe number with which we can be sure in single step?
gmat-admin's picture

Unfortunately, there's no safe number that will always eliminate 4 of the 5 answer choices in 1 step.
That's the downside of testing numbers.

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