Expert: Ahmed Salami - 5/18/2004

Question
what is the product of the following series?
(X-A)x(X-B)x(X-C)x(X-D)...(X-Z)

Answer
Hi,
Assuming 26 factors and letting
f(x)= (x-a)(x-b)(x-c).........(x-z)
this is an expression whose roots are a,b,c,......,z
By the theory of equations, expansion gives
f(x)= x^26 - Ax^25 + Bx^24 - Cx^23 + ......... -Yx + Z
(note the alternation of signs)
where A = sum of the roots i.e a+b+c+.....+z
     B = sum of product of the roots, two at a time i.e (ab+ac+ad+.....az+bc+bd+.....+bz+cd+.......+yz)
     C = sum of product of the roots, three at a time i.e (abc+abd+.........+xyz)
     .
     .
     .
     Z = abcde......z
I am sure you can see the sequence.

But generally speaking, if there are n factors in that order
    A is the sum of n terms
    B is the sum of nC2 (read 'n combination 2') terms
    C is the sum of nC3 terms
And so on.

I hope i have really been helpful.
You can always ask more.

NB: nCr = n!/r!(n-r)!
where n!= n(n-1)(n-2)(n-3)......(3)(2)(1)