Simple Algebra

Algebra Level 2

Which of the following numbers is not a solution of this equation: ( x − 1 ) ( x − 2 ) ( x − 3 ) + ( x − 1 ) ( x − 2 ) ( x − 4 ) + ( x − 1 ) ( x − 2 ) ( x + 10 ) = 0 (x-1)(x-2)(x-3)+(x-1)(x-2)(x-4)+(x-1)(x-2)(x+10)=0

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2 3 1 -1

Jaime Andrés by Jaime Andrés , 21, Spain

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1 solution

Daniel Liu
Aug 30, 2015

Note that the expression factorizes as ( x − 1 ) ( x − 2 ) ( x − 3 + x − 4 + x + 10 ) = ( x − 1 ) ( x − 2 ) ( 3 x + 3 ) = 3 ( x − 1 ) ( x − 2 ) ( x + 1 ) \begin{aligned} (x-1)(x-2)(x-3+x-4+x+10)&=(x-1)(x-2)(3x+3)\\ &=3(x-1)(x-2)(x+1) \end{aligned} which has roots x = 1 , 2 , − 1 x=1, 2, -1 .

This leaves 3 \boxed{3} as not being a solution.

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