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If (x-4) / (x^2 - 5x + 6) can be expande...

If `(x-4) / (x^2 - 5x + 6)` can be expanded in the ascending powers of x, then the coefficient of `x^3`

A

` ( - 73 )/(648) `

B

`(73)/(648) `

C

`(71)/(648) `

D

` ( - 71)/(648)`

Text Solution

Verified by Experts

` (x - 4 ) / ( x ^ 2 - 5x + 6 ) `
` = (2 ) /((x - 2 )) - (1)/((x - 3 )) `
` therefore (2) /((x - 2 )) - (1)/((x-3)) = (2)/((-2) (1 - (x ) /(2)) ) + (1)/(3(1 - (x)/(3)) `
= ` (1 ) /(3) (1 - (x )/(3)) ^( -1) - (1 - (x)/(2)) ^(-1) `
` = (1)/(3) [ 1 + (x)/(3) + ((x)/(3)) ^ 2 + (x/3) ^ 3 + ... ] `
` - [ 1 + (x )/(2) + (x/2) ^2 + (x/2 ) ^ 3 + ... ] `
` therefore ` Co - efficient of ` x ^ 3 ` in above expansion is
` (1)/(3) (1/3) ^3 - (1/2) ^3 = (1)/(81) - (1)/(8) = ( -73)/(648) `
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