2022 AMC 12B Problem 20

Problem 20 of 25HarderAlgebra Number Theory

Let

P(x)

be a polynomial with rational coefficients such that when

P(x)

is divided by the polynomial

x^2 + x + 1,

the remainder is

x + 2,

and when

P(x)

is divided by the polynomial

x^2 + 1,

the remainder is

2x + 1.

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?

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Solution

No linear polynomial can have the two different remainders. If

P(x)=ax^2+bx+c

were quadratic, comparing its remainders would give both

b-a=1,\ c-a=2

and

b=2,\ c-a=1,

a contradiction. Thus the least possible degree is

3.

Write

P(x) = (x + 2)

+ (x^2 + x + 1)(px + q),

which has remainder

x + 2

upon division by

x^2 + x + 1.

Reducing modulo

x^2 + 1

(so

x^2 \equiv -1

) gives remainder

(q + 1)x + (2 - p).

Setting this equal to

2x + 1

gives

q = 1

and

p = 1.

Then

P(x) = x^3 + 2x^2 + 3x + 3,

and the sum of the squares of the coefficients is

1 + 4 + 9 + 9 = 23.

Thus, the correct answer is E.