2022 AMC 10B Problems/Problem 21
Problem
Let be a polynomial with rational coefficients such that when is divided by the polynomial , the remainder is , and when is divided by the polynomial , 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?
Solution 1 (Experimentation)
Given that all the answer choices and coefficients are integers, we hope that has positive integer coefficients.
Throughout this solution, we will express all polynomials in base . E.g. .
We are given:
.
We add and to each side and balance respectively:
We make the units digits equal:
We now notice that:
.
Therefore , , and . is the minimal degree of since there is no way to influence the ‘s digit in when is an integer. The desired sum is
P.S. The 4 computational steps can be deduced through quick experimentation.
~ numerophile
Video Solutions
~ ThePuzzlr
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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