Question

If \(\frac{3x-1}{(x-1)(x-2)(x-3)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3}\), then the values of (A, B, C) are

• A. (1, -5, 4)
• B. (1, 5, 4)
• C. (4, 5, 1)
• D. (1, 4, 5)

Hint:

The "cover-up" method is extremely fast for finding coefficients in partial fractions when the denominator has distinct linear factors. To find the coefficient A for a term \(\frac{A}{x-a}\), "cover-up" the \((x-a)\) factor in the original fraction's denominator and substitute \(x=a\) into the rest of the expression. For example, to find B: cover up \((x-2)\) in \(\frac{3x-1}{(x-1)(x-2)(x-3)}\) to get \(\frac{3x-1}{(x-1)(x-3)}\), and substitute \(x=2\): \(\frac{3(2)-1}{(2-1)(2-3)} = \frac{5}{(1)(-1)} = -5\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the values of the constants A, B, and C in the partial fraction decomposition of the given rational function.

Step 2: Key Formula or Approach (Alternate Method):
Use the cover-up method directly on the original fraction without expanding the identity first. This is the fastest method for distinct linear factors.

Step 3: Detailed Explanation:
Given: (3x-1)/[(x-1)(x-2)(x-3)] = A/(x-1) + B/(x-2) + C/(x-3). For A: Cover (x-1) in denominator and substitute x=1 in remaining expression: A = (3(1)-1)/[(1-2)(1-3)] = 2/[(-1)(-2)] = 2/2 = 1. For B: Cover (x-2) and substitute x=2: B = (3(2)-1)/[(2-1)(2-3)] = 5/[(1)(-1)] = 5/(-1) = -5. For C: Cover (x-3) and substitute x=3: C = (3(3)-1)/[(3-1)(3-2)] = 8/[(2)(1)] = 8/2 = 4.

Step 4: Final Answer:
The values are A=1, B=-5, and C=4. So, (A, B, C) = (1, -5, 4).