Factor x^6 - 64 over Real numbers?
1 Answer
Feb 10, 2018
Explanation:
The difference of squares identity can be written:
a^2-b^2 = (a-b)(a+b)
The difference of cubes identity can be written:
a^3-b^3 = (a-b)(a^2+ab+b^2)
The sum of cubes identity can be written:
a^3+b^3 = (a+b)(a^2-ab+b^2)
Hence we find:
x^6-64 = (x^3)^2-8^2
color(white)(x^6-64) = (x^3-8)(x^3+8)
color(white)(x^6-64) = (x^3-2^3)(x^3+2^3)
color(white)(x^6-64) = (x-2)(x^2+2x+4)(x+2)(x^2-2x+4)
Notes
It is interesting to see what happens if you use the difference of cubes identity first:
x^6-64 = (x^2)^3-4^3
color(white)(x^6-64) = (x^2-4)((x^2)^2+4x^2+4^2)
color(white)(x^6-64) = (x^2-2^2)((x^2+4)^2-(2x)^2)
color(white)(x^6-64) = (x-2)(x+2)((x^2+4)-2x)((x^2+4)+2x)
color(white)(x^6-64) = (x-2)(x+2)(x^2-2x+4)(x^2+2x+4)
