I need to know how to factor the following expression:
x^6 - y^6
The teacher said that the answer is (x+y)(x-y)(x^4+x^2y^2+y^4)
I need to know HOW to factor this.
Does it have anything to do with making x^6 = (x^2)^3??
To factor the expression x^6 - y^6, you can use a special factorization formula called the difference of squares. In this case, x^6 - y^6 can be written as:
(x^2)^3 - (y^2)^3
Now, you can see that this expression follows the pattern of the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). If you let a = x^2 and b = y^2, you can rewrite the expression as:
(x^2)^3 - (y^2)^3 = (x^2 - y^2)(x^4 + x^2y^2 + y^4)
At this point, you have successfully factored the expression into (x^2 - y^2)(x^4 + x^2y^2 + y^4). However, you can go one step further to factor (x^2 - y^2) using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).
Using this formula with a = x and b = y, you can further simplify (x^2 - y^2) to:
(x^2 - y^2) = (x + y)(x - y)
Therefore, the complete factorization of the expression x^6 - y^6 is:
x^6 - y^6 = (x + y)(x - y)(x^4 + x^2y^2 + y^4)
So, to factor the expression x^6 - y^6, you first use the difference of cubes formula to factor out (x^2 - y^2) as (x + y)(x - y). Then, you have the factorization (x + y)(x - y)(x^4 + x^2y^2 + y^4).