How do you expand ${(u - v)}^{3}$ $(u-v)^3$ ?

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Answer 1 · 2016-07-09T12:24:44

$= u^{3} - 3 u^{2} v + 3 u v^{2} - v^{3}$ $= u^3 - 3u^2v + 3uv^2 - v^3$

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 $x = u, y = -v and n = 3$

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

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

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

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