What is the coefficient of \(x^3\) in this expression?

\((x^4 + x^3 + x^2 + x + 1)^4\)

thank you to whoever helps :)

lokiisnotdead Apr 23, 2020

#2**+1 **

According to WolframAlpha it is 20

https://www.wolframalpha.com/input/?i=%28x%5E4+%2Bx%5E3+%2Bx%5E2%2Bx%2B1%29%5E4

ElectricPavlov Apr 23, 2020

#4**+2 **

The coefficients of the final (multiplied-out) terms form tetrahedral numbers: 1, 4, 10, 20, 34, 56, ...

So, the coefficient of the constant term (the 1) and the x^{16}-term are both 1.

The coefficient of the x-term and the x^{15}-term are both 4.

The coefficient of the x^{2}-term and the x^{14}-term are both 10.

The coefficient of the x^{3}-term and the x^{13}-term are boh 20

Etc.

To find the n^{th}-term of the tetrahedral numbers, you can use this formula: n · (n + 1)·(n + 2) / 6

geno3141 Apr 23, 2020