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This problem trains for: SAT-II, AMC-10, GMAT.

A computer program outputs polynomial expressions based on their roots. If the roots are given, what additional data does the program need in order to figure out a single matching polynomial?

According to the fundamental theorem of algebra, any n-th degree polynomial:


has n complex roots:


and can be factored as:


One can see that, to specify a unique polynomial with degree n, n+1 constants are needed: either all the n+1 coefficients, or the n roots and one additional constant which is the leading coefficient an.

The program also needs a value for the leading coefficient.

We do not need the degree of the polynomial since it is related to the number of constants that form the input data.