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555-605 (Medium)|   Algebra|      
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Bunuel
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If a^3 a^2 a + 1 = 0, what is the product of all possible values o 15 Apr 2022, 11:30
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

35% (medium)

Question Stats:

64% (01:25) correct 36% (01:08) wrong based on 73 sessions

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If \(a^3 − a^2 − a + 1 = 0\), what is the product of all possible values of a?

A. -1
B. 0
C. 1
D. 2
E. 4
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A
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BrentGMATPrepNow
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If a^3 a^2 a + 1 = 0, what is the product of all possible values o 15 Apr 2022, 12:23
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Bunuel
If \(a^3 − a^2 − a + 1 = 0\), what is the product of all possible values of a?

A. -1
B. 0
C. 1
D. 2
E. 4
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We can factor the left side in parts.

Given: a³ − a² − a + 1 = 0

Factor in parts: a²(a − 1) - 1(a - 1) = 0

Combine to get: (a² - 1)(a − 1) = 0

Since the first expression (and brackets) is a difference of squares, we can factor it as follows: (a + 1)(a - 1)(a - 1) = 0

There are two unique solutions: a = 1 and a = -1, which means the product of all possible values of a = (1)(-1) = -1

Answer: A
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If a^3 a^2 a + 1 = 0, what is the product of all possible values o 02 Jun 2022, 12:04
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BrentGMATPrepNow
Bunuel
If \(a^3 − a^2 − a + 1 = 0\), what is the product of all possible values of a?

A. -1
B. 0
C. 1
D. 2
E. 4

We can factor the left side in parts.

Given: a³ − a² − a + 1 = 0

Factor in parts: a²(a − 1) - 1(a - 1) = 0

Combine to get: (a² - 1)(a − 1) = 0

Since the first expression (and brackets) is a difference of squares, we can factor it as follows: (a + 1)(a - 1)(a - 1) = 0

There are two unique solutions: a = 1 and a = -1, which means the product of all possible values of a = (1)(-1) = -1

Answer: A
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Hey Brent, really nice solution here. I answered this question slightly differently by testing different integer values (starting with 0, then -1,1, then 2,-2 etc), so I realised that only 1 and -1 seemed to work, hence picked A.

I like your solution and was wondering if you have any tips on how to be able to develop an ability to spot opportunities to factorise like this? I would really like to try and develop such foresight, especially the part where (after factoring by parts), you noted that you could place a 1 outside the second component and make it -1(a-1) and then use difference of two squares, how can I learn / develop the ability to spot opportunities to do this?
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If a^3 a^2 a + 1 = 0, what is the product of all possible values o 03 Jun 2022, 06:46
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Hey Brent, really nice solution here. I answered this question slightly differently by testing different integer values (starting with 0, then -1,1, then 2,-2 etc), so I realised that only 1 and -1 seemed to work, hence picked A.

I like your solution and was wondering if you have any tips on how to be able to develop an ability to spot opportunities to factorise like this? I would really like to try and develop such foresight, especially the part where (after factoring by parts), you noted that you could place a 1 outside the second component and make it -1(a-1) and then use difference of two squares, how can I learn / develop the ability to spot opportunities to do this?
Show more

When an algebraic expression consists of four terms (as it does here), we should be watching out for the possibility of factoring in parts.

Here are a few questions to practice with:
- https://gmatclub.com/forum/the-area-of- ... l#p1995500
- https://gmatclub.com/forum/if-u-and-v-a ... l#p1837587
- https://gmatclub.com/forum/what-is-the- ... l#p2456284
- https://gmatclub.com/forum/if-mx-qy-nx- ... l#p2446623
- (super tricky!): https://gmatclub.com/forum/m28-184512.html#p2038496
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