Bunuel wrote:
If \(x \neq 2\), then \(\frac{3x^2(x-2)-x+2}{x-2}\)
(A) \(3x^2 - x + 2\)
(B) \(3x^2 + 1\)
(C) \(3x^2\)
(D) \(3x^2 - 1\)
(E) \(3x^2 - 2\)
STRATEGY: Upon reading any GMAT Problem Solving question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test the answer choices for equivalency.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we can also try to simplify the expression so that it matches one of the answer choices.
I think testing for equivalency will be faster, so I'll go with that...Key concept: If two expressions are equivalent, they must evaluate to the same value for every possible value of x.
For example, since the expression 2x + 3x is equivalent to the expression 5x, the two expressions will evaluate to the same number for every value of x.
So, if x = 7, the expression 2x + 3x = 2(7) + 3(7) = 14 + 21 = 35, and the expression 5x = 5(7) = 35Let's test an easy value like \(x = 0\).
Plug \(x = 0\) into the given expression to get: \(\frac{3x^2(x-2)-x+2}{x-2}=\frac{3(0)^2(0-2)-0+2}{0-2} = \frac{2}{-2}= -1\)
Now we'll plug \(x = 0\) into the five answer choices see which one(s) evaluate(s) to \(-1\)
(A) \(3(0)^2 - 0 + 2 = 2\). ELIMINATE
(B) \(3(0)^2 + 1 = 1\). ELIMINATE
(C) \(3(0)^2 = 0\). ELIMINATE
(D) \(3(0)^2 - 1 = -1\)
GREAT! Keep. (E) \(3(0)^2 - 2 = -2\). ELIMINATE
Answer: D