2250.11 – A Nice Rational Equation

Let

35x−29x2−3x+2=ax−1+bx−2\frac{35x - 29}{x^2 - 3x+2} = \frac{a}{x-1} + \frac{b}{x-2}

be an identity in xx. Then abab equals:

  1. -246
  2. -210
  3. -29
  4. 210
  5. 246
Solution

What an unattractive problem! Well, let's do it:

35x−29x2−3x+2=35x−29(x−1)(x−2)=ax−1+bx−2=a(x−2)+b(x−1)(x−1)(x−2)=ax−2a+bx−b(x−1)(x−2)=(a+b)x−(2a+b))(x−1)(x−2)=35x−29(x−1)(x−2).\begin{aligned} \frac{35x - 29}{x^2 - 3x+2} &=& \frac{35x - 29}{(x - 1)(x-2)} \\ &=& \frac{a}{x-1} + \frac{b}{x-2} \\ &=& \frac{a (x-2)+b(x-1)}{(x - 1)(x-2)} \\ &=& \frac{ax-2a+bx-b}{(x - 1)(x-2)} \\ &=& \frac{(a+b)x-(2a +b))}{(x - 1)(x-2)} \\ &=& \frac{35x - 29}{(x - 1)(x-2)}. \end{aligned}

So,

a+b=352a+b=29,\begin{aligned} a + b &=& 35 \\ 2a+b &=& 29, \end{aligned}

from which it follows that a=−6a = -6, b=41b = 41, and ab=−246.ab = -246. The answer is a.