Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
One way to attack this problem is to factor the given expression: x3 – 4x5 = x3(1 – 4x2)
Notice that 1 – 4x2 is a difference of squares. This part of the expression factors into (1 – 2x)(1 + 2x)
So the whole expression becomes x3(1 – 2x)(1 + 2x).
To trace the sign changes of the whole expression, track what happens to each part of the product.
x3 is negative when x is less than zero, but it’s positive when x is greater than zero.
(1 – 2x) is positive when x is less than ½, but it’s negative when x is greater than ½. (Careful about the sign change.)
(1 + 2x) is negative when x is less than –½, but it’s positive when x is greater than –½.
Now you have three “break points” where signs change: –½, 0, ½. This means that you have four regions to examine. You might set up a quick table to take care of the cases, or you can just talk your way through them.
1) x is less than –½: first term is negative, second is positive, third is negative, so the product is positive.
2) x is between –½ and 0: first term is still negative, second is still positive, but third is now positive. So the product is negative.
3) x is between 0 and ½: first term is now positive. Second is still positive, third is positive, so the product is positive.
4) x is greater than ½: first term is positive, but now second term is negative. Third is still positive, so the product is negative.
Cases 2 and 4 give us a negative product. You can also test numbers, of course, but given the high powers, you might not want to raise fractions to these powers.
The correct answer is C.