2250.11 – A Nice Rational Equation

Let

35x−29x2−3x+2=ax−1+bx−2\dfrac{35x - 29}{x^2 - 3x+2} = \dfrac{a}{x-1} + \dfrac{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} \dfrac{35x - 29}{x^2 - 3x+2} &= \dfrac{35x - 29}{(x - 1)(x-2)} \\[1em]&= \dfrac{a}{x-1} + \dfrac{b}{x-2} \\[1em]&= \dfrac{a (x-2)+b(x-1)}{(x - 1)(x-2)} \\[1em]&= \dfrac{ax-2a+bx-b}{(x - 1)(x-2)} \\[1em]&= \dfrac{(a+b)x-(2a +b))}{(x - 1)(x-2)} \\[1em]&= \dfrac{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=−246ab = -246. The answer is a.