Math Counts Question #16

Algebra Level 2

For all x x in the domain of the function x + 1 x 3 − x \dfrac {x+1}{x^3-x} , the function is equivalent to:

1 x 3 − 1 x \frac 1{x^3} - \frac 1x 1 x 2 − 1 \frac 1{x^2-1} 1 x 2 − x \frac 1{x^2-x} 1 x 2 − 1 x 3 \frac 1{x^2} - \frac 1{x^3} 1 x 3 \frac 1{x^3}

Eric Zhu by Eric Zhu , 15, USA

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1 solution

x + 1 x 3 − x = x + 1 x ( x 2 − 1 ) = x + 1 x ( x − 1 ) ( x + 1 ) = 1 x ( x − 1 ) = 1 x 2 − x \dfrac {x+1}{x^3-x} = \dfrac {x+1}{x(x^2-1)} = \dfrac {\color{#D61F06}x+1}{x(x-1)\color{#D61F06}(x+1)} = \dfrac 1{x(x-1)} = \boxed{\dfrac 1{x^2-x}}

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