Question

How to factor x^6 - y^6

Answer 1

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

Answer 2

In your case you can say, \[\huge {\color{black} {(x^{3})^2-(y ^{3})^2} }\]

Answer 3

\(\bf x^6-y^6\implies (x^3)^2-(y^3)^2\qquad \textit{recall that }\ a^2-b^2 = (a-b)(a+b)\)

Answer 4

... so... what would that give you as factors?

Answer 5

@Zunz

Answer 6

couldnt this be factored more with the sum of cubes equation?

Answer 7

yes given that \(\bf a^3+b^3 = (a+b)(a^2-ab+b^2) \qquad a^3-b^3 = (a-b)(a^2+ab+b^2)\)

Answer 8

so wouldnt the factors be (x + y)(x - y)(x 2 + xy + y 2)(x 2 - xy + y 2)

Answer 9

yeap

Answer 10

my main question is how do you know when to use (x3)2−(y3)2

Answer 11

depends on the exponent, if you can squeeze a "2" out of it, or more than one 2, then you do

Answer 12

for example something like \(\bf x^{15}-y^{15}\) wouldn't give us a "2" from the exponent, so, can't use it there

Answer 13

so in that case it'd end up like \(\bf x^{15}-y^{15}\implies (x^5)^3-(y^5)^3\implies (x^5-y^5)[(x^5)^2+x^5y^5+(y^5)^2]\)