Question

Factor `36x^3+12x^2-72x-24` ?

Answer 1

`36x^3+12x^2-72x-24 = 12(x-sqrt(2))(x+sqrt(2))(3x+1)`

Explanation:

Note that the ratio of the `1`st and `2`nd terms is the same as that between the `3`rd and `4`th terms. So this cubic will factor by grouping. Also note that all of the coefficients are divisible by `12`, so we can separate that out as a factor first:

`36x^3+12x^2-72x-24 = 12(3x^3+x^2-6x-2)`

`color(white)(36x^3+12x^2-72x-24) = 12((3x^3+x^2)-(6x+2))`

`color(white)(36x^3+12x^2-72x-24) = 12(x^2(3x+1)-2(3x+1))`

`color(white)(36x^3+12x^2-72x-24) = 12(x^2-2)(3x+1)`

`color(white)(36x^3+12x^2-72x-24) = 12(x^2-(sqrt(2))^2)(3x+1)`

`color(white)(36x^3+12x^2-72x-24) = 12(x-sqrt(2))(x+sqrt(2))(3x+1)`