AMC 10B 2022 (Problem 21)
Let be a polynomial with rational coefficients such that when is divided by , the remainder is , and when is divided by , the remainder is . There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
Try small degrees for . Use Remainder Theorem for quadratics.
If : impossible.
If : contradiction between the two remainders.
If : contradiction between the two remainders.
If : let . Use and to derive equations.
From the system: , , , .
Solve to obtain , , , .
. Sum of squares of coefficients: . (E) 23