How do I use the the binomial theorem to expand #(v - u)^6#?

1 Answer
Mar 5, 2018

Answer:

#color(blue)(v^6-6v^5u+15v^4u^2-20v^3u^3+15v^2u^4-6vu^5+u^6)#

Explanation:

For a binomial expansion:

#(x+y)^n# we have:

#((n),(r))x^(n-r)y^r#

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

Where:

#((n),(r))=color(white)(0)^n C_(r)=(n!)/(r!(n-r)!)#

Beginning with #r=0#

#((6),(0))v^6(-u)^0+((6),(1))v^5(-u)^1+((6),(2))v^4(-u)^2#

#+((6),(3))v^3(-u)^3+((6),(4))v^2(-u)^4+((6),(5))v^1(-u)^5#

#((6),(6))v^0(-u)^6#

Calculating #((n),(r))#

#(1)v^6(-u)^0+(6)v^5(-u)^1+(15)v^4(-u)^2#

#+(20)v^3(-u)^3+(15)v^2(-u)^4+(6)v^1(-u)^5#

#(1)v^0(-u)^6#

Expand brackets and simplify:

#color(blue)(v^6-6v^5u+15v^4u^2-20v^3u^3+15v^2u^4-6vu^5+u^6)#

To make things easier we can use the following:

#color(white)(0)^nC_(r)=color(white)(0)^nC_(n-r)#

And:

#(-u)^n# is negative for odd powers and positive for even powers.