Not So Big!

Algebra Level 3

(xb)(xc)(xd)(ab)(ac)(ad)+(xc)(xd)(xa)(bc)(bd)(ba)+(xd)(xa)(xb)(cd)(ca)(cb)+(xa)(xb)(xc)(da)(db)(dc) = ? \begin{aligned} & \dfrac{(x-b)(x-c)(x-d)}{(a-b)(a-c)(a-d)} \\+ & \dfrac{(x-c)(x-d)(x-a)}{(b-c)(b-d)(b-a)} \\ +& \dfrac{(x-d)(x-a)(x-b)}{(c-d)(c-a)(c-b)} \\ +& \dfrac{(x-a)(x-b)(x-c)}{(d-a)(d-b)(d-c)} \end{aligned} ~=~?

Hint: An equation of nthn^ \text{th} degree with more than nn roots is an identity. This is studied extensively in the wiki: The method of undetermined coefficients.

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