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

1 Answer
Mar 5, 2018

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.