Really HARD Factoring problem, Help???

Question

Factor Completely:

(a+b)^6 - (a-b)^6

Help me!!!
P.s. ^6 means raised to the 6th power

Answer 1

(a+b)^6 - (a-b)^6

Factor as a difference of squares
x² - y² = (x + y)(x - y)

[(a+b)^3 + (a-b)^3][(a+b)^3 - (a-b)^3]

Factor as sum and difference of cubes
x³ + y³ = (x + y)(x² - xy +y²)
x³ - y³ = (x - y)(x² + xy +y²)

[a + b + a - b][(a+b)^2 - (a+b)(a-b) + (a-b)^2]
[a + b - a + b][(a+b)^2 + (a+b)(a-b) + (a-b)^2]

Simplify

[2a][a² + 2ab + b² - a² + b² + a² - 2ab + b²]
[2b][a² + 2ab + b² + a² - b² + a² - 2ab + b²]

[2a][a² + 3b²][2b][3a² + b²]

4ab(a² + 3b²)(3a² + b²)

Answer 2

If you have a graphing calculator, like a TI-83, I suggest graphing the above. You can then find all the zeros, which in turn can give you the roots. For example, your sister's problem gives the answers of x = -3, -1.5, -.8 and 1 if y = 0. This gives the factors of (x + 3) (2x + 3) (5x + 4) (x - 1). For a problem like this, without using a calculator, you'd have to plug in values for x and see where the equation equals zero. Once you find a root (aka a factor as well), you'd have to do synthetic division. I don't know how far along you are mathematically...

Answer 3

Hint: Factor it as the difference of 2 squares.

Answer 4

((a+b)^3 +(a+b)^3)((a+b)^3-(a+b)^3)