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Find the coefficient of $x^9$ in the power series of each of these functions.

a) $(x^3+x^5+x^6).(x^3+x^4).(x+x^2+x^3+x^4+⋯)$

b) $(1+x+x^2)^3$
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for 1st case:

the most efficient method would be ::

coefficient of x^9 in (x3+x5+x6).(x3+x4).(x+x2+x3+x4+⋯).

                              =x3(1+x2+x3) x3(1+x) x(1+x+x2+x3....)

                               =x7(1+x2+x3)(1+x)(1+x+x2+x3......)

coefficient of x^2 in (1+x2+x3)(1+x)(1+x+x2+x3......)

                          =(1+x2)(1+x)(1+x+x2)     removing polynomial >=3.

                         = (1+x2) (1+x+x2+x+x2+x4)

                         =(1+x2) (1+2x+2x2)       removing polynomial >=3

                          =(1+2x +2x2 +x2+2x3+2x4)

                          =(1+2x+3x2)

therefore coefficient of x2 = 3

means coefficient of x9 = 3.


for case 2: 

(we dont need to solve that as (1+x+x2)3 will never have x9 as coefficient and it would be 0 as the maximum degree of polynomial would be X6 only).

and if we solve it :

we have (1+x+x^2)^3 = (1-x3)3 / (1-x) 3 = (1-x3)3 / (1--x) -3

                                  = (1-x3)3 (n+r-1)C(n-1) Xr 

                                 = (1-x9-3x3(1-x3)) (2+r)C(2) Xr

                               = (1-x9-3x3-3x6) (2+r)C2 Xr

                               = 11C2 -2C0 -3*(8C2) + 3(5C2)

                               = 55-1- 28*3  +30

                               =0 answer.


 

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