+0  
 
+1
47
5
avatar+365 

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

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

 

thank you to whoever helps :)

 Apr 23, 2020
 #1
avatar+928 
0

1 is the coefficient.

 Apr 23, 2020
 #2
avatar+23518 
+1

According to WolframAlpha  it is 20

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

 Apr 23, 2020
 #3
avatar+365 
+1

Yeah, 20 is right, but I'm confused as to why...

lokiisnotdead  Apr 23, 2020
edited by lokiisnotdead  Apr 23, 2020
 #4
avatar+20767 
+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 x16-term are both 1.

The coefficient of the x-term and the x15-term are both 4.

The coefficient of the x2-term and the x14-term are both 10.

The coefficient of the x3-term and the x13-term are boh 20 

Etc.

 

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

 Apr 23, 2020
 #5
avatar+365 
+1

wow! thank you so much geno! that makes so much more sense now!

lokiisnotdead  Apr 23, 2020

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