500 followers problem!
Let be a monic cubic polynomial with , and . Evaluate , where represents the Gamma function.
The answer is 322.
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1 solution
Because F(x) is monic we can say:
F ( x ) = x 3 + a x 2 + b x + c
Define a quartic polynomial P(x) such that:
P ( x ) = F ( x ) − x 4 P ( x ) = − x 4 + x 3 + a x 2 + b x + c
Note that 1, 2, and 3 are roots of P(x). Let us call the 4th unknown root r.By vieta's formulas
1 + 2 + 3 + r = − − 1 1 = 1 r = − 5
Knowing its four roots we may rewrite P(x) as
P ( x ) = − ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x + 5 )
Now solve for P(4) and use it to solve for F(4)
P ( 4 ) = − ( 3 ) ( 2 ) ( 1 ) ( 9 ) = − 5 4 P ( 4 ) = F ( 4 ) − 2 5 6 − 5 4 = F ( 4 ) − 2 5 6 2 0 2 = F ( 4 )
Finally, calculate Γ ( 6 ) and add it to F(4) to get the answer
Γ ( 6 ) = ( 6 − 1 ) ! = 5 ! = 1 2 0 F ( 4 ) + Γ ( 6 ) = 2 0 2 + 1 2 0 = 3 2 2