Question

How do you factor `6x^4-9x^2+3`?

Answer 1

`6x^4-9x^2+3 = 3(sqrt(2)x-1)(sqrt(2)x+1)(x-1)(x+1)`

Explanation:

The difference of squares identity can be written:

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

We will use this a couple of times.

Before that, separate out the common scalar factor `3` then treat as a quadratic in `x^2`...

`6x^4-9x^2+3 = 3(2x^4-3x^2+1)`

`color(white)(6x^4-9x^2+3) = 3(2(x^2)^2-3(x^2)+1)`

`color(white)(6x^4-9x^2+3) = 3(2x^2-1)(x^2-1)`

`color(white)(6x^4-9x^2+3) = 3((sqrt(2)x)^2-1^2)(x^2-1^2)`

`color(white)(6x^4-9x^2+3) = 3(sqrt(2)x-1)(sqrt(2)x+1)(x-1)(x+1)`