Question

How do you factor #y=4x^3+12x^2-16x-48 #?

Answer 1

`x^3+3x^2-4x-12 = 0`

you need to find numbers such as :

`x_1 + x_2 + x_3 = -3`

`x_1x_2x_3 = 12`

Because for every monic polynomial of degree `n` you have

Sum of all root = `-b`

Product of all root = `(-1)^nk`

For this take all the divisor of `+-12`

`-12 ; -6 ; -4 ; -3 ; -2 ; -1 ; 1 ; 2 ; 3 ; 4 ; 6 ; 12`

Sum must be negative so one root is negative, BUT, product is positive so, at least two roots are negative

you find

`x_1 = -3 ; x_2 = -2 ; x_3 = 2`

Of course this method work if coefficient are integers

So :

(x+3)(x+2)(x-2)