How do you find the constant term of a polynomial using binomial expansion?

Example: ( (a + b^(1/3) - c^(1/2) )^7

Question

How do you find the constant term of a polynomial using binomial expansion?

Example: ( (a + b^(1/3) - c^(1/2) )^7

mhchen · Answer

$$(a + b^{1/3} - c^{1/2})^{7}$$

mhchen · Answer

If someone just types "Any Ideas?" >_< 

I know how to find the constant term in $$(a - \frac{ 1 }{ b })^{c}$$

Which is just making the powers equal. 

I've never learned how to apply that to 3 parts.

phi · Answer

binomial expansion

the "bi" means 2
so it does not really apply to polynomials  (poly means "many") bigger than 2

mhchen · Answer

Alright...I guess trinomial expansion or polynomial expansion... whatever

phi · Answer

yes, wiki has a short page on it https://en.wikipedia.org/wiki/Trinomial_expansion

mhchen · Answer

Okay so according to that article, it looks like each coefficient is given with that sigma thing. Do you expect me to find every term using that sigma?

phi · Answer

it looks painful!

the sigma means add up the terms
each term has a coefficient  n!/(i! j! k!)
and a, b, c to the i,j,k powers
i,j and k take on all combinations of numbers (from 0 to n) such that all 3 add up to n

holsteremission · Answer

Do you have a more concrete example? If $a,b,c$ are all constants, then the expansion itself is a constant.

Maybe you mean something more along the lines of $(x+y+1)^n$?

mhchen · Answer

Nah, I got a harder problem but I just simplified it down to get the overall concept on how to do it. I got it though.

I was told to find the coefficient of x^2yz term, and I just researched the trinomial expansion basically.