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What is the largest possible value of. . .

by Vincen » Tue Dec 12, 2017 10:51 am
What is the largest possible value of the following expression (x+2)(3−x)(x−3)(2x+4)(2+x)^2 ?

A. -846
B. 54
C. 12
D. 0
E. can't be determined

The OA is D.

I thought the correct answer was E. How can I determine the largest possible value? Experts, I would be thankful if you help me here.
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Answer

by Vincen » Tue Dec 12, 2017 11:40 am
I still cannot know how to find the largest value.
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by EconomistGMATTutor » Tue Dec 12, 2017 11:43 am
Hello Vincen.

We have to rewrite the expression to know the possible largest value.

$$(x+2)(3-x)(x-3)(2x+4)(2+x)^2\ =\ -2\left(x+2\right)\left(x-3\right)\left(x-3\right)\left(x+2\right)\left(x+2\right)^2$$ That implies that the original expression is equal to $$-2\left(x-3\right)^2\left(x+2\right)^4\ =\ negative\left(positive\right)\left(positive\right)\le0,\ \ \ \forall x\in\mathbb{R}.$$ So, the possible largest value is 0, and it is obtained when x=-2 or x=3.

This is why the correct answer is D.

I hope this explanation may help you.

I'm available if you'd like a follow-up.

Regards.
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