8/29/2014 If the graph of the rational function, f(x) = (2b – 2ax + bx + 2x² – ax² + x³)/(x² – x) does not have any vertical asymptotes, what is the value of a?

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8/29/2014 Solution. Since (2b – 2ax + bx + 2x² – ax² + x³)/(x² – x) does not have any V.A.s, (2b – 2ax + bx + 2x² – ax² + x³) must be a multiple of (x² – x). Now,

2b – 2ax + bx + 2x² – ax² + x³ can be factored as 2(b – ax + x²) + x(b – ax + x²)

= (x + 2)(b – ax + x²). This implies (b – ax + x²) should be a multiple of (x² – x) so that there would be no V.A. → b = 0. Rewriting (0 – ax + x²) = (x² – ax ) = x(x – a), clearly, a = 1.