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Question 18

Let $$\frac{3x - 25}{x^2 - 5x + 6} = \frac{a}{x - 3} + \frac{b}{x - 2}$$ be an identity on $$x$$. Then the numerical value of $$b - a$$ is


Correct Answer: 35

Solution

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Since $$x^2 - 5x + 6 = (x - 2)(x - 3)$$, clearing denominators gives $$3x - 25 = a(x - 2) + b(x - 3)$$. Putting $$x = 3$$ gives $$-16 = a$$, and putting $$x = 2$$ gives $$-19 = -b$$, so $$b = 19$$. Hence $$b - a = 19 + 16 = 35$$.

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