Question

How do you expand the binomial `(2x-y^3)^7` using the binomial theorem?

Answer 1

`128x^7-448x^6y^3+672x^5y^6-560x^4y^9+280x^3y^12-84x^2y^15`
`+14xy^18-y^21`

Explanation:

For the expansion of `(x+y)^n` we have:

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

Where:

`((n),(r))=color(white)(0)^nC_(r)=(n!)/((r!(n-r)!)`

`(2x-y^3)^7`

`((7),(0))(2x)^7(-y^3)^0+((7),(1))(2x)^6(-y^3)^1+((7),(2))(2x)^5(-y^3)^2`

`+((7),(3))(2x)^4(-y^3)^3+((7),(4))(2x)^3(-y^3)^4+((7),(5))(2x)^2(-y^3)^5`

`+((7),(6))(2x)^1(-y^3)^6+((7),(7))(2x)^0(-y^3)^7`

Using `\ \ \ (n!)/((r!(n-r)!)`

`128x^7-448x^6y^3+672x^5y^6-560x^4y^9+280x^3y^12-84x^2y^15`
`+14xy^18-y^21`