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2019/20 AHSMC Part 1 Problem 4

Problem 4 of 16

The polynomial P(x)=(2x3−1)673P(x) = (2x^3 - 1)^{673} can be expanded as (2x3−1)673=a2019x2019+a2018x2018+⋯+a1x+a0. (2x^3 - 1)^{673} = a_{2019} x^{2019} + a_{2018} x^{2018} + \cdots + a_1 x + a_0. Then, the sum a2019+⋯+a1+a0a_{2019} + \cdots + a_1 + a_0 is

Answer choices

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Solution

Setting x=1x = 1 we get a2019+⋯+a1+a0=(2−1)673=1. a_{2019} + \cdots + a_1 + a_0 = (2-1)^{673} = 1. The answer is (B).