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प्रश्न
Resolve the following rational expressions into partial fractions
`x/((x - 1)^3`
उत्तर
`x/(x - 1)^3 = "A"/(x - 1) + "B"/(x - 1)^2 + "C"/(x - 1)^3`
`x/(x - 1)^3 = ("A"(x - 1)^2 + "B"(x - 1) + "C")/(x - 1)^3`
x = A(x – 1)2 + B(x – 1) + C ......(1)
Put x = 1 in equation (1)
⇒ 1 = A(1 – 1)2 + B(1 – 1) + C
1 = 0 + 0 + C
⇒ C = 1
In equation (1), equating the coefficient of x2 on both sides
0 = A ⇒ A = 0
Put x = 0 in equation (1) ⇒ 0 = A(0 – 1)2 + B(0 – 1) + C
⇒ 0 = A – B + C
0 = 0 – B + 1
⇒ B = 1
∴ The required partial fraction is `x/(x - 1)^3 = 1/(x - 1)^2 + 1/(x - 1)^3`
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