2017 2017

Algebra Level 3

if x = 2017 + 1 x=\sqrt{2017}+1 , find the value of x 3 − ( 2 + 2017 ) x 2 + ( 1 + 2 2017 ) x − 2017 x^3-(2+\sqrt{2017})x^2+(1+2\sqrt{2017})x-\sqrt{2017} ?


The answer is 2017.

Moch Abdul Aziz by Moch Abdul Aziz , 24, Indonesia

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1 solution

Chew-Seong Cheong
Jun 27, 2018

X = x 3 − ( 2 + 2017 ) x 2 + ( 1 + 2 2017 ) x − 2017 Given that x = 2017 + 1 ⟹ 2017 = x − 1 = x 3 − ( x + 1 ) x 2 + ( 2 x − 1 ) x − x + 1 = x 3 − x 3 − x 2 + 2 x 2 − x − x + 1 = x 2 − 2 x + 1 = ( x − 1 ) 2 = ( 2017 ) 2 = 2017 \begin{aligned} X & = x^3 - (2+\sqrt{2017})x^2 + (1+2\sqrt{2017})x - \sqrt{2017} & \small \color{#3D99F6} \text{Given that }x=\sqrt{2017}+1 \implies \sqrt{2017} = x-1 \\ & = x^3 - (x+1)x^2 + (2x-1)x - x+1 \\ & = x^3 - x^3 - x^2 + 2x^2 - x - x + 1 \\ & = x^2 - 2x + 1 \\ & = (x-1)^2 \\ & = (\sqrt{2017})^2 \\ & = \boxed{2017} \end{aligned}

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