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Let P(x) be a polynomial such that p(x)-...

Let P(x) be a polynomial such that p(x)-p'(x)= `x^(n)`, where n is a positive integer. Then P(0) equals-

A

n!

B

(n-1)!

C

`(1)/(n!)`

D

`(1)/((n-1)!)`

Text Solution

Verified by Experts

The correct Answer is:
A
`Let P(x)=a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-1)+ . . . +a_(n)`
`P^(')(x)=na_(0)x^(n-1)+(n-1)a_(1)+ . . .+a_(n-1)`
`P(x)P^(')(x)na_(0)x^(n)+(a_(2)-(n-1)a_(1))x^(n-2)+ . . .+(a_(n)-a_(n-1))`
given `P(x)-P^(')=a_(0)x^(n)`.
so that
`a_(1)-na_(0)=0" "(a_(1))/(a_(0))=n`
`a_(2)-(n-1)a_(1)=0" "(a_(2))/(a_(1))=(n-1)`
`a_(n)-a_(n-1)=0" "(a_(n))/(a_(n-1))=1`
`P(0)=a_(n)=((a_(n))/(a_(n-1)))((a_(n-1))/(a_(n-2))). . .((a_(1))/(a_(0)))`
`=1xx2xx3xx . . .xxn`
=n!
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