Please solve the question in the attachment.............

this is quite an easy ques and can be done by assuming the polynomial as p(x)

then the given eqn becomes

 

d/dx(e^x*p(x))=x^n*e^x

or e^x(p(x)+p’(x))=x^n*e^x

or p(x)+p’(x)=x^n

obviously it is easy to find find p’(x) as it is a polynomial. now you can compare the coefficients of x^r on both sides of the eqn so as to form a recurrence relation from which ar is easily deduced.

in fact a_r/a_r-1 = n+1 – r, which is precisely true for ar= nPr (permutations)

but nPr= n!/(n – r)!

hence, a_r= nPr= n!/(n – r)!