Factoring: [tex]8x^{24}-27y^6[/tex]
Theory : A difference of two perfect cubes, [tex]a^3-b^3[/tex] can be factored into
[tex](a-b)*(a^2+ab+b^2)[/tex]
Proof : [tex](a-b)*(a^2+ab+b^2)=[/tex]
[tex]a^3+a^2b+ab^2-ba^2-b^2a-b^3=[/tex]
[tex]a^3+(a^2b-ba^2)+(ab^2-b^2a)-b^3=[/tex]
[tex]a^3+0+0+b^3=[/tex]
[tex]a^3+b^3[/tex]
Check : 8 is the cube of 2
Check : 27 is the cube of 3
Check : [tex]x^{24}[/tex] is the cube of [tex]x^8[/tex]
Check : [tex]y^6[/tex] is the cube of [tex]y^2[/tex]
Factorization is :
[tex](2x^8 - 3y^2)*(4x^{16} + 6x^8y^2 + 9y^4)[/tex]
