Question

How do you factor the expression `4x^4-6x^2+2`?

Answer 1

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

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

Explanation:

Given:

`4x^4-6x^2+2`

Note that the sum of the coefficients is zero, i.e. `4-6+2=0`.

So we can tell that `x=+-1` are zeros and `(x-1)(x+1) = x^2-1` is a factor.

Also all of the terms are divisible by `2`, so we could separate that out as a factor, but given what we find, it's probably better to leave it in there until the end:

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

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

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

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