Question

How do you factor `3x^4 - 6x^3 - 4x^2 + 8x`?

Answer 1

`3x^4-6x^3-4x^2+8x = x(3x^2-4)(x-2)`

`color(white)(3x^4-6x^3-4x^2+8x) = 3x(x-(2sqrt(3))/3)(x+(2sqrt(3))/3)(x-2)`

Explanation:

We can separate out the common factor `x` then factor by grouping as follows:

`3x^4-6x^3-4x^2+8x = x(3x^3-6x^2-4x+8)`

`color(white)(3x^4-6x^3-4x^2+8x) = x((3x^3-6x^2)-(4x-8))`

`color(white)(3x^4-6x^3-4x^2+8x) = x(3x^2(x-2)-4(x-2))`

`color(white)(3x^4-6x^3-4x^2+8x) = x(3x^2-4)(x-2)`

`color(white)(3x^4-6x^3-4x^2+8x) = 3x(x^2-4/3)(x-2)`

`color(white)(3x^4-6x^3-4x^2+8x) = 3x(x^2-((2sqrt(3))/3)^2)(x-2)`

`color(white)(3x^4-6x^3-4x^2+8x) = 3x(x-(2sqrt(3))/3)(x+(2sqrt(3))/3)(x-2)`