Alphabetical multiplication

Algebra Level pending

What is the sum of the numerical coefficients of the powers of x x when ( x − a ) ( x − b ) ( x − c ) . . . ( x − z ) (x-a)(x-b)(x-c)...(x-z) is expanded?

NB: The numerical coefficients of both 2 a x 2 2ax^2 and 2 ( a + b ) x 2 2(a+b)x^2 would be 2 2 in this case.


The answer is 0.

Jake Lyons by Jake Lyons , 20, New Zealand

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1 solution

Jake Lyons
Jan 30, 2017

Through close inspection of the factorised expression, one can write it as:

( x − a ) . . . ( x − x ) ( x − y ) ( x − z ) (x-a)...(x-x)(x-y)(x-z)

Notice that x − x = 0 x-x=0 .

Therefore: ( x − a ) . . . ( x − w ) 0 ( x − y ) ( x − z ) = 0 (x-a)...(x-w)0(x-y)(x-z)=0

So all the possible powers of x x will have a coefficient of 0 0 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Haha, Nice Question, I Didnt Expected this, but when i inserted the answer, it matched!

Md Zuhair - 4 years, 4 months ago

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