PUMaC 2021 · 代数（B 组） · 第 5 题

PUMaC 2021 — Algebra (Division B) — Problem 5

Let f ( x ) = 1 + 2 x + 3 x + 4 x + 5 x and let ζ = e = cos + i sin . Find the value of 5 5 the following expression: 2 3 4 f ( ζ ) f ( ζ ) f ( ζ ) f ( ζ ) .

解析

Let f ( x ) = 1 + 2 x + 3 x + 4 x + 5 x and let ζ = e = cos + i sin . Find the value of 5 5 the following expression: 2 3 4 f ( ζ ) f ( ζ ) f ( ζ ) f ( ζ ) . Proposed by: Michael Gintz Answer: 125 Write this as the product 5 5 5 4 ( x − 1) + ( x − x ) + . . . + ( x − x ) f ( x ) = x − 1 6 5 5 x − 6 x + 1 = , 2 ( x − 1) 5 which for these terms will be equal to . Thus taking this for each of our four multiplicands, ( x − 1) 2 3 4 the denominator becomes ((1 − ζ )(1 − ζ )(1 − ζ )(1 − ζ ) = 5, so our answer is 125.