AMC 10B 2022 (Problem 21)
PolynomialsHard
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 examples for the degree of .
If , then
When is divided by , the remainder is , which cannot equal .
When is divided by , the remainder is , which cannot equal .
If , then
Since the degree of is less than the degrees of both divisors, the remainder in each division is .
For division by ,
For division by ,
This is impossible.
Since the degree of is less than the degrees of both divisors, the remainder in each division is .
For division by ,
For division by ,
This is impossible.
If , let
For division by , by Polynomial Remainders,
Therefore,
For division by , by Polynomial Remainders,
Therefore,
But cannot be both and , so this is impossible.
For division by , by Polynomial Remainders,
Therefore,
For division by , by Polynomial Remainders,
Therefore,
But cannot be both and , so this is impossible.
If , let
For division by , by Polynomial Remainders,
and
Thus,
Therefore,
For division by , by Polynomial Remainders,
Thus,
Therefore,
Hence,
Thus,
Using ,
Therefore,
For division by , by Polynomial Remainders,
and
Thus,
Therefore,
For division by , by Polynomial Remainders,
Thus,
Therefore,
Hence,
Thus,
Using ,
Therefore,
(E)
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