How do you expand #(u-v)^3#?

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Answer 1 · 2016-07-09T12:24:44

#= u^3 - 3u^2v + 3uv^2 - v^3#

You can use the Binomial Theorem, which is given by:

#(x+y)^n = sum_(k=0)^n ((n),(k)) x^(n-k)y^k#

where #((n),(k)) = (n!)/(k!(n-k)!)#

with #x = u, y = -v and n = 3#

#therefore (u-v)^3 = sum_(k=0)^3 ((3),(k))u^(3-k)(-v)^k#

#=((3),(0))u^3(-v)^0 + ((3),(1))u^2(-v)^1 + ((3),(2))u^1(-v)^2 + ((3),(3))u^0(-v)^3#

#= u^3 - 3u^2v + 3uv^2 - v^3#