Polynomial identity ($P(x)=Q(x)$)

A polynomial identity is equality of two polynomials for every value of the variables, rather than an equation imposed only at selected roots. Over the real or complex numbers, this is equivalent to equality of corresponding coefficients: their difference, if nonzero, has only finitely many roots in one variable. The binomial theorem is an example. Coefficient comparison transfers a polynomial product identity into finite sums of binomial coefficients.