Factor:
x^8 + 2x^4 y^4 + 9y^8
To factor the given expression, x^8 + 2x^4y^4 + 9y^8, we can use the formula for sum of cubes:
a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Notice that the given expression, x^8 + 2x^4y^4 + 9y^8, can be rearranged as:
(x^8 + 2x^4y^4 + y^8) + 8y^8
= (x^8 + 2x^4y^4 + y^8) + (3y^4)^2
= (x^4 + y^4)^2 + (3y^4)^2.
Now, we can rewrite it as the sum of two cubes:
(a^3 + b^3) = (a + b)(a^2 - ab + b^2),
where a = (x^4), and b = (3y^4).
Substituting these values, we have:
(x^8 + 2x^4y^4 + 9y^8) = [(x^4) + (3y^4)][(x^4)^2 - (x^4)(3y^4) + (3y^4)^2].
Simplifying further, we get:
(x^8 + 2x^4y^4 + 9y^8) = (x^4 + 3y^4)(x^4^2 - 3x^4y^4 + 9y^8).
Therefore, the factored form of the expression x^8 + 2x^4y^4 + 9y^8 is (x^4 + 3y^4)(x^4^2 - 3x^4y^4 + 9y^8).