(x-A)(x-B)(x-D)(x-E)....(x-Z)?

Question

A little help with this math problem, help me get on the right track.

Simplify the following polynomial..
(x-A)(x-B)(x-D)(x-E).....(x-Z)

Answer 1

the answer is 0
at point you will have (x-x) that is zero .. so whole thing multiplied with 0 will yield zero ... cheers :)

Answer 2

It definitely won't be simple if by simplification, you mean multiplied out.

Alright, let's work this out from first principle (or try to). First, if we have:

P_1 = (x-a)(x-b) = x² - (a+b)x + ab
P_2 = P_1(x-c) = x³ - (a+b+c)x² + (ab+ac+bc)x - abc
P_3 = P_2(x-d) = x^4 - (a+b+c+d)x³ + (ab+ac+ad+bc+bd+cd)x² - (abc+abd+acd+bcd)x + abcd

Now, it seems we have a pattern. It isn't hard to notice the alternating of the signs between negative and positive. We can expect this to continue, especially as it is what happens in the binomial theorem as well, for some (x-a)^n.

We can see the pattern. For instance, when calculating P_4, we can see that the first term is obviously x^5 with a coefficient of 1. For the second term, we see that it is -(a+b+c+d) + (coefficient of previous (1st) term)(-e). Because the constant term -e must be multiplied by the term with a degree one higher than the one multiplied by x.

If we carry this out for all the terms, we can see that the eventual coefficients become enumerations of the possible multiplications of all the variables taken (n-1) at a time when n is the current term. It is horribly difficult for me to explain, especially since my communicating skills are poor, but try to work with it and see how it is.

In any case, the entire expanded form is just wow!. There will be 27 terms with degrees from 26 to 0. The term with degree 13 will have an amazing 26C13 = 10,400,600 terms in its coefficient alone!.

It's impossible to enumerate out. In any case, the pattern is:

x^26 - (a+b+...+z)x^25 + (ab+ac+...+bc+bd+...+yz)x^24 - (abc+abd+...+xyz)x^23 + ... - (abc...yz+acd...yz+...+bcd...yz)x + abcd...xyz

I think I'm correct, if someone could check my work please :-D

@rkbhat26,

Wow, you got me there. I didn't even think of that even though I heard of this question before. I think it was the capitals that threw me off. Are x and X the same?

Answer 3

Looks pretty simplified to me in current form. Else, you're just multiplying it out and that's a mess!

Answer 4

one of the factors will be x-x which will be equal to 0 and when one of the factors is 0 the entire product will be 0.so the answer is zero

Answer 5

Pascal's Triangle?

Answer 6

ok if you had a problem like (x-7)+a
and the instructions were x is 9 and a is 3
put them in there place

Answer 7

0 because x-x is zero and anything times 0 is 0 check ya later ♥

Answer 8

I am a senior in college. And still don't know the correct answer. If thats makes you feel better.

Answer 9

That is not the complete problem.

Answer 10

What are you supposed to do with ths?