Question

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

Answer 1

`color(green)((x+y)^5 = x^5 + 5x^4y + 10 x^3y^2 + 10 x^2y^3 + 5xy^4 + y^5)`

Explanation:

`(x+y)^5 = 5C0 x^5 y^0 + 5C1 x^4 y^1 + 5C2 x^3 y^2 + 5C3 x^2 y^3 + 5C4 x y^4 + 5C5 x^0 y^5`

`= x^5 + 5 x^4y + ((5*4)/(1*2)) *x^3y^2 + ((5*4*3)/(1*2*3)) x^2 y^3 + ((5*4*3*2)/(1*2*3*4)) xy^4 + y^5`

`color(green)( = x^5 + 5x^4y + 10 x^3y^2 + 10 x^2y^3 + 5xy^4 + y^5)`

Answer 2

`color(blue)((x+y)^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5)`

Explanation:

Alternate Method using Pascal's Triangle,

1 5 10 10 5 1

`color(blue)((x+y)^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5)`