Question

How do you factor `x^6+10x^3+25`?

Answer 1

`x^6+10x^3+25 = (x^3+5)^2 = (x+root(3)(5))^2(x^2-(root(3)(5))x+root(3)(25))^2`

Explanation:

The sum of cubes identity can be written:

`a^3+b^3=(a+b)(a^2-ab+b^2)`

Use this with `a=x` and `b=root(3)(5)` as follows:

`x^6+10x^3+25 = (x^3)^2+2(5)(x^3)+(5)^2`

`color(white)(x^6+10x^3+25) = (x^3+5)^2`

`color(white)(x^6+10x^3+25) = (x^3+(root(3)(5))^3)^2`

`color(white)(x^6+10x^3+25) = ((x+root(3)(5))(x^2-x(root(3)(5))+(root(3)(5))^2))^2`

`color(white)(x^6+10x^3+25) = (x+root(3)(5))^2(x^2-(root(3)(5))x+root(3)(25))^2`