When all's unknown

Calculus Level 5

f(x)=ax3+bx2+cx+d \large{ f(x) = ax^3 + bx^2 + cx + d}

The Taylor Series expansion of f(x) f(x) about x=a x= a is

d6(xa)3+(3a+b+2d)(xa)2+(3a2+c+6d)(xa)\frac{d}{6}(x- a)^3 + (3a + b + 2d)(x- a)^2 + (3a^2 + c + 6d)(x - a)

Determine the product abcdabcd, where aa, bb, cc, and dd are distinct, nonzero, coprime (not necessarily pairwise) integers.

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