Zeroing out, part 2

Algebra Level 2

Let x x be a real number. Consider the function

f ( x ) = x 3 − 1 x 2 + x − 2 f(x) = \frac{x^3 - 1}{x^2 + x - 2}

How many real zeroes does it have?

2 3 0 1

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Feb 4, 2016

Factoring, the function becomes ( x − 1 ) ( x 2 + x + 1 ) / ( x − 1 ) ( x + 2 ) (x-1)(x^2 + x + 1) / (x-1)(x+2)

The only possible zero would be at 1, since the quadratic has no real zeroes, but at 1 the function is of the form 0/0, which is indeterminate. It has a removable singularity there, not a zero.

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