One key technique for expression simplification GMAT Math questions is to figure out a ‘smart’ value to plug in.

Let’s start with an example:

**What is the simplification of (x ^{2} + 5x + 2)(y^{2} + 2x + 3)(z^{2} + 4x + 10)?**

**A. (60 + 214x + 58xy ^{2} + 5xy^{2}z^{2} + 19xz^{2} + 206x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2})**

**B. (60 + 214x + 58xy ^{2} + 5xy^{2}z^{2} + 18xz^{2} + 207x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2})**

**C. (60 + 214x + 58xy ^{2} + 5xy^{2}z^{2} + 40xz^{2} + 206x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2})**

**D. (40 + 214x + 58xy ^{2} + 5xy^{2}z^{2} + 19xz^{2} + 206x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2})**

**E. (13 + 214x + 58xy ^{2} + 5xy^{2}z^{2} + 19xz^{2} + 206x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2})**

Readers are encouraged to give the above problem a shot. Challenge yourself to not spend more than 45 sec – 1 min 🙂

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There is absolutely no way we are multiplying and simplifying the original equation (x^{2} + 5x + 2)(y^{2} + 2x + 3)(z^{2} + 4x + 10) as it would take a ton of time.

**What we want to do is to exclude options by plugging real numbers. What sort of numbers do we plug in? We want to use the easiest numbers like 0 and 1 which when raised to a power remain the same and do not need additional calculation.**

**We start with 0**

(x^{2} + 5x + 2)(y^{2} + 2x + 3)(z^{2} + 4x + 10) when x, y and z = 0 becomes

2 x 3 x 10 = 60

Plugging into the answer options

Option (D) and (E) are out since they yield 40 and 13 when x, y and z are plugged with 0

It would be great if we could eliminate all but 1 option by plugging 0. Q50+ questions are seldom this easy but we are ready for the challenge 🙂

**Let’s go with 1 now.**

(x^{2} + 5x + 2)(y^{2} + 2x + 3)(z^{2} + 4x + 10) when x, y and z = 1 becomes

8 x 6 x 15 = 720

Plugging into the answer options (A), (B) and (C)

Only (A) and (B) satisfy this.

Now, we are left with (A) and (B) and 0 and 1s are unable to differentiate them

A powerful tool to tackle such as situation is to find the ‘algebraic difference’ between (A) and (B)

Which is essentially (A) – (B) = (60 + 214x + 58xy^{2} + 5xy^{2}z^{2} + 19xz^{2} + 206x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2}) ** – ** (60 + 214x + 58xy^{2} + 5xy^{2}z^{2} + 18xz^{2} + 207x^{2} + 30x^{2}y^{2} + x^{2}y^{2}z^{2} + 13x^{2}z^{2} + 72x^{3} + 4x^{3}y^{2} + 2x^{3}z^{2} + 8x^{4} + 20y^{2} + 2y^{2}z^{2} + 6z^{2})

= xz^{2} – x^{2} (Notice the subtraction is straightforward as most of the terms are the same – this is by design as the GMAT want to make it hard to differentiate between the two options by making all terms almost the same – we leverage this to our benefit)

The next steps is to figure out what simple value would make the difference ‘non zero’

Plugging xz^{2} – x^{2} with 0 leads to 0

But plugging with x = 1, y = 0 z = 0 yields -1

**The set of values which lead to a non-zero differential is what we use as the next set of values to test with.**

**x = 1, z = 0 and y = 0**

(x^{2} + 5x + 2)(y^{2} + 2x + 3)(z^{2} + 4x + 10)

(8) x (5) x (14) = 560

Plugging into option (A) we get 560 and into B we get 561 (in fact, we didn’t have to plug into B – as soon as 1 option works out we are assured the other option will give a different value because of the differential)

We go with **option (A)**

A recap of the algorithm is below