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Question
Resolve the following rational expressions into partial fractions
`(2x^2 + 5x - 11)/(x^2 + 2x - 3)`
Solution
`(2x^2 + 5x - 11)/(x^2 + 2x - 3)`
Since the degree of the numerator is equal to the degree of the denominator divide the numerator by the denominator
`(2x^2 + 5x - 11)/(x^2 + 2x - 3) = 2 + (x - 5)/(x^2 + 2x - 3)` ......(1)
Consider `(x - 5)/(x^2 + 2x - 3)`
`(x - 5)/(x^2 + 2x - 3) = (x - 5)/(x^2 + 3x - x - 3)`
= `(x - 5)/(x(x + 3) - 1(x + 3))`
`(x - 5)/(x^2 + 2x - 3) = (x - 5)/((x - 1) (x + 3))`
`(x - 5)/(x^2 + 2x - 3) = "A"/(x - 1) + "B"/(x + 3)` ......(2)
`(x - 5)/(x^2 + 2x - 3) = ("A"(x + 3) + "B"(x - 1))/((x - 1) (x + 3))`
x – 5 = A(x + 3) + B(x – 1) ......(3)
Put x = 1 in equation (3)
1 – 5 = A(1 + 3) + B(1 – 1)
– 4 = 4A + 0
⇒ A = – 1
Put x = – 3 in equation (3)
– 3 – 5 = A(– 3 + 3) + B(– 3 – 1)
– 8 = 0 – 4B
⇒ B = 2
Substituting the values of A and B in equation (2) we have
`(x - 5)/(x^2 + 2x - 3) = (-1)/(x - 1) + 2/(x + 3)`
∴ The required partial fraction is
`(2x^2 + 5x - 11)/(x^2 + 2x - 3) = 2 + 2/(x + 3) - 1/(x - 1)`
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