Question

How do you use the binomial theorem to expand and simplify the expression `(2x-y)^5`?

Answer 1

`=-y^5+2(5(y^4x-4x^2y^3+8x^3y^2-8yx^4)+16x^5)`

Explanation:

The binomial expansion of `(ay+bx)^n=(ay)^n+n(ay)^(n-1)bx+((n),(2))(ay)^(n-2)(bx)^2+cdots+((n),(n-2))(ay)^2(bx)^(n-2)+n(ay)(bx)^(n-1)+(bx)^n`

So, `(-y+2x)^5=(-y)^5+5(-y)^4(2x)+((5),(2))(-y)^3(2x)^2+((5),(3))(-y)^2(2x)^3+5(-y)(2x)^4+(2x)^5`
`=-y^5+10y^4x-40x^2y^3+80x^3y^2-80yx^4+32x^5`
`=-y^5+10(y^4x-4x^2y^3+8x^3y^2-8yx^4)+32x^5`
`=-y^5+10(y^4x-4x^2y^3+8x^3y^2-8yx^4)+32x^5`
`=-y^5+2(5(y^4x-4x^2y^3+8x^3y^2-8yx^4)+16x^5)`