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प्रश्न
Resolve the following rational expressions into partial fractions
`(3x + 1)/((x - 2)(x + 1))`
उत्तर
`(3x + 1)/((x - 2)(x + 1)) = "A"/(x - 2) + "B"/(x + 1)`
`(3x + 1)/((x - 2)(x + 1)) = ("A"(x + 1) + "B"(x - 2))/((x - 2)(x + 1))`
3x + 1 = A(x + 1) + B(x – 2) ......(1)
Put x = 2 in equation (1)
3(2) + 1 = A(2 + 1) + B(2 – 2)
6 + 1 = 3A + 0
⇒ A = `7/3`
Put x = – 1 in equation (1)
3(– 1) + 1 = A(– 1 + 1) + B(– 1 – 2)
– 3 + 1 = A × 0 – 3B
– 2 = 0 – 3B
⇒ B = `2/3`
∴ The required partial fractions are
`(3x + 1)/((x - 2)(x + 1)) = (7/3)/(x - 2) + (2/3)/(x + 1)`
`(3x + 1)/((x - 2)(x + 1)) = 7/(3(x - 2)) + 2/(3(x + 1))`
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