What is the coefficient of $x$ in $(x^4 + x^3 + x^2 + x + 1)^4$?

Question

What is the coefficient of  in ?

Answer 1

Let a=x^4+x^3+x^2 and b=x+1, then the expression becomes (a+b)^4. The only term which includes x^1 is b^4 when (a+b)^4. The last term of the expansion is b^4=(x+1)^4=x^4+4x^3+6x^2+4x+1. The coefficient of the x term is therefore 4.

Another way to solve this is to put x=10, then we have 11111^4=123454321^2. We only need to take the lower end of this number and square it: 21^2=441; 321^2=103041; 4321^2=18671041, etc.

Since x=10 we just need the tens digit which is 4 in each case.

Answer 2

What is the coefficient of  in

​Replace  (x^4 + x^3 + x^2) with m,  and (x + 1)​ with u, then we have

​(m + u)^4

Expanding this gives,

​m^4 + 4.m^3.u + 6.m^2.u^2 + 4.m.u^3 + u^4

​every term above, when expanded, will have all constituent terms with powers of x greater than 2, except for the last term, u^4.

​expanding u^4 = (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1

​Coefficient of x in the expansion of the original expression is: 4