2022 AMC 12B Problems/Problem 20
Contents
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
It is easy to see that has a degree of at least 2.
Suppose that it has degree , so let
. Then comparing coefficients of
gives
, and comparing coefficients of
gives
, a contradiction.
Now suppose it has degree . Let
. Equating coefficients of
gives
, so
.
Equating coefficients of gives
, so
and
.
Now equating coefficients of gives
and hence
. Hence
.
Then, we equate coefficients of to get
, so
.
Hence, and the sum of the squares of coefficients is
, and we're done!
Video Solution by OmegaLearn Using Polynomial Remainders
~ pi_is_3.14
See Also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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