Rooting out the answer
Let A x 4 + B x 3 + C x 2 + D x + E = 0 be an equation with integer coefficients such that A is positive, g cd ( A , B , C , D , E ) = 1 , and the roots are 2 , − 2 , 3 , − 3 .
Find A + B + C + D + E .
The answer is 2.
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1 solution
Moderator note:
Simple standard approach.
At the risk of being picky, the equation − x 4 + 5 x 2 − 6 = 0 would also satisfy all the given requirements, but would result in the sum of the coefficients being − 2 instead of 2 . Perhaps it would be an idea to also state that A > 0 so that the posted solution is uniquely correct.
You can also do it by writing :
F ( x ) = ( x ± 2 ) ( x ± 3 ) And what we are asked to find is F ( 1 ) which is ( 1 ± 2 ) ( 1 ± 3 ) = ( − 1 ) ( − 2 ) = 2
For roots of 2 and − 2 , the equation is x 2 − 2 = 0 .
For roots of 3 and − 3 the equation is x 2 − 3 = 0
So the equation we want is ( x 2 − 2 ) × ( x 2 − 3 ) = 0 , or ( x 4 − 5 x 2 + 6 ) = 0
A = 1, B = 0, C = -5, D = 0, E = 6, so A + B+ C + D + E = 2.