**8/29/2014****
**If
the graph of the rational function, f(x) = (2b – 2ax + bx + 2x^{2}
– ax^{2
}+
x^{3})/(x^{2}
– x)
does not have any vertical asymptotes, what is the value of *a*?

**Scroll
down for the solution.**

**8/29/2014
**Solution.
Since
(2b – 2ax + bx + 2x^{2}
– ax^{2
}+
x^{3})/(x^{2}
– x)
does not have any V.A.s, (2b – 2ax + bx + 2x^{2}
– ax^{2
}+
x^{3})
must be a multiple of (x^{2}
– x).
Now,

2b
– 2ax + bx + 2x^{2}
– ax^{2
}+
x^{3
}can
be factored as 2(b – ax + x^{2})
+ x(b – ax + x^{2})

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