To solve the problem of finding a cubic polynomial (f(x)) with leading coefficient 1 that passes through the points ((1,2)), ((2,4)), ((3,6)), we can use the following reasoning:

Key Insight:

We need a cubic polynomial where (f(x) - 2x) is divisible by ((x-1)(x-2)(x-3)) (since (f(1)-2(1)=0), (f(2)-2(2)=0), (f(3)-2(3)=0)). For a cubic polynomial with leading coefficient 1, the divr ((x-1)(x-2)(x-3)) (a cubic) is exactly the term to add to (2x).

Solution:

[f(x) = (x-1)(x-2)(x-3) + 2x]

Expanding this gives:
[f(x) = x^3 - 6x^2 + 13x - 6]

Answer: (\boxed{x^3 - 6x^2 + 13x - 6}) (or the factored form (\boxed{(x-1)(x-2)(x-3)+2x}) if preferred, but expanded form is more standard).

But since the assistant's response was the factored form, the final answer is:

(\boxed{(x-1)(x-2)(x-3)+2x})

作者声明:本文包含人工智能生成内容。