Question

How do you factor the expression `3x - 9x^3`?

Answer 1

`3x-9x^3`

`= 3x(1-3x^2)`

`= 3x(1-sqrt(3)x)(1+sqrt(3)x)`

`= x(sqrt(3)-3x)(sqrt(3)+3x)`

Explanation:

We can use the difference of squares identity:

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

to factor with irrational coefficients.

First notice that both terms are divisible by `3x`, so separate that out as a factor:

`3x-9x^3`

`= 3x(1-3x^2)`

`= 3x(1^2-(sqrt(3)x)^2)`

`= 3x(1-sqrt(3)x)(1+sqrt(3)x)`

Or if you prefer, split that factor of `3` between the two binomials:

`3x-9x^3`

`= x(3-9x^2)`

`= x((sqrt(3))^2-(3x)^2)`

`= x(sqrt(3)-3x)(sqrt(3)+3x)`