請高手幫忙(x+y)^6-(x-y)^6

If you want to factorize the function, the following is the answer:

(x + y)^6 - (x - y)^6
= [(x + y)^3]^2 - [(x - y)^3]^2
= [(x + y)^3 + (x - y)^3][(x + y)^3 - (x - y)^3]
= {[(x + y) + (x - y)][(x + y)^2 - (x + y)(x - y) + (x - y)^2]} {[(x + y) - (x - y)][(x + y)^2 + (x + y)(x - y) + (x - y)^2]}
= 2x[x^2 + 2xy + y^2 - (x^2 - y^2) + x^2 - 2xy + y^2] X 2y[x^2 + 2xy + y^2 + (x^2 - y^2) + x^2
- 2xy + y^2]
= 2x(x^2 + 3y^2) X 2y(3x^2 + y^2)
= 4xy(x^2 + 3y^2)(3x^2 + y^2)

If you want to expand it, then you can do one more step:

4xy(x^2 + 3y^2)(3x^2 + y^2)
= 4xy(3x^4 + 10(x^2)(y^2) + 3y^4)
= 12x^5y + 40x^3y^3 + 12xy^5

If you need to use the binomial theorem, just tell me, thanks!

吾識簡單啲既方法, 睇住先啦

(x+y)^6
=x^6+6(x^5)y+15(x^4)(y^2)+20(x^3)( y^3)+15(x^2)( y^4)+6x( y^5)+y^6

(x-y)^6
=x^6-6(x^5)y+15(x^4)(y^2)-20(x^3)( y^3)+15(x^2)( y^4)-6x( y^5)+y^6

(x+y)^6-(x-y)^6
=6(x^5)y+20(x^3)( y^3)+6x( y^5)-[-6(x^5)y-20(x^3)( y^3) -6x( y^5)]
=12(x^5)y+40(x^3)( y^3)+12x( y^5)

應該啱嫁啦^^