What is the largest possible value of the following expression

Question

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

Bunuel · Accepted Answer

Factor -1 from the second multiple (3-x) and 2 from the 4th multiple (2x+4):

(x+2)(3-x)(x-3)(2x+4)(2+x)^2=-2(x+2)(x-3)(x-3)(x+2)(2+x)^2= 
 =-2(2+x)^4*(x-3)^2=-(some \ nonnegative \ value) . So, the max value of the expression is 0.

Answer: D.

JohanH · Answer

(3-x) is negative if x>3
(x-3) is negative if x<3

The balance terms will always be positive as they reduce to 2(x+2) to the power of 4 which is always positive.

So for all x values except for x=3 the total expression yields a negative number.
If x=3 it is multiplication with one term being zero, so the answer is zero

Zero is the largest possible value - so D.

mvictor · Answer

if x is positive, then the whole expression will result in a negative number, since we have 4 positive numbers and 1 negative number. If x is 0, again same thing. To maximize the expression, we need to make sure that at least one of the expressions results in 0. this can be done only if x is =-2 or x=3. So the maximum possible result is 0.

jasimuddin · Answer

as (3-x)(x-3) is here to make It largest, max value of x=3 . For x=3 largest value of the expression = 0
ans:D

Fdambro294 · Answer

(X + 2) (3 - X) (X - 3) (2X + 4) (2 + X)^2

We are trying to maximize this expression. Start by asking: is there any way to have the result be Positive?

Case 1: If we pick any negative value less than < -3

(3 - X) will be positive and (2 + X)^2 will be positive

However the other 3 factors will be negative.

Since we have an odd number of negative Factors (3 negative and 2 positive ) ———> result will be negative

Case 2: if we pick a value from -3 to 0, non inclusive

We will either end up with 1 factor negative (x - 3)

Or we will have 3 factors negative (x + 2) and (x - 3) and (2x + 4)

Either way, again we have an ODD number of negative Factors ———> so Product will be Negative

Case 3: if we choose an X value from 0 to 3, non inclusive

Every factor will be positive EXCEPT ——-> (X - 3)

Since again we have ONE negative factor and the rest positive (an ODD number of negative factors) ———> the product will end up being Negative

Case 4: X is greater than 3

In this case every factor will be positive except one factor again

This time it is the OPPOSITE Term of case 3 ———-> (3 - X)

Again, we will always end up with a negative quotient

No matter which numbers we pick, for all real values of X, we can NEVER get a positive result

Thus, we can make one of the factors = 0 (let X = 3), leading to the MAX Product we can get:

0

Posted from my mobile device

rvgmat12 · Answer

The expression becomes -2*(x+2)^4 * (x-3)^2

Max value is 0 when x=3

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the largest possible value of the following expression

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the largest possible value of the following expression

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards