Question

How do you factor `24x^3-6x^2+8x-2`?

Answer 1

`24x^3-6x^2+8x-2 = 2(3x^2+1)(4x-1)`

Explanation:

Notice that the ratio betweeen the first and second terms is the same as that between the third and fourth terms. So this cubic will factor by grouping:

`24x^3-6x^2+8x-2 = (24x^3-6x^2)+(8x-2)`

`color(white)(24x^3-6x^2+8x-2) = 6x^2(4x-1)+2(4x-1)`

`color(white)(24x^3-6x^2+8x-2) = (6x^2+2)(4x-1)`

`color(white)(24x^3-6x^2+8x-2) = 2(3x^2+1)(4x-1)`

The remaining quadratic factor has no linear factors with Real coefficients, since `3x^2+1 >= 1` for all Real values of `x`.