Question regarding what appears to be an identity

Question

This is an MCQ we were posed in school recently (I hope you don't mind elementary stuff):

What is $(x-a)(x-b)(x-c)...(x-z)$ ? Options:

$0$

$1$

$2$

$(x^n)-(abcdef...z)$

@mixedmath Surely it did, or else none of the choices is correct (not that I particularly like the "solution" in that case).
Oh yes, it did. I'm so sorry. But the last option was still $x^n - (abc...z)$ AFAIK. I lost the test paper :(
@SohamChowdhury The fact that $n$ or $z$ actually makes a difference is something you should think about.

Mike · Accepted Answer · 2012-09-27 03:13:17Z

up vote 7 down vote accepted

There is a very basic trick to this problem. It all comes down to a single term (if that's the proper word for it...).

The only real hint I can give is $x$ is a letter between $a$ and $z$...

answered Sep 27 '12 at 3:13

Mike
3,4251316

+1, Very cool! Also, Thank you @DonAntonio for pointing where I went wrong. – Emmad Kareem Sep 27 '12 at 4:11

1

Ah, I see... $(x-a)(x-b)..(x-x)..(x-z)$ where $(x-x)$ is 0. Nice! – Soham Chowdhury Sep 27 '12 at 4:22

Gone · Answer 2 · 2012-09-27 03:57:20Z

up vote 3 down vote

Hint $\ $ What is $\rm\ (24-1)(24-2)(24-3)\cdots (24-26)\ $ ?

And what is $\rm\,(x_{24}\!-x_1)(x_{24}\!-x_2)(x_{24}\!-x_3)\cdots (x_{24}\!-x_{26})\ $ ?

edited Sep 27 '12 at 3:57

user242

answered Sep 27 '12 at 3:50

Gone
1

@Thanks for the hint. Very good point. – Emmad Kareem Sep 27 '12 at 4:08

2 Answers