Question

Question #66e6f

Answer 1

`f(x)=3x^3-5x^2-12x+20=(x+2)(x-2)(x-5/3)`

Explanation:

The rational root theorem basically tells us what the possible zeroes could be of a polynomial function. To use the rational root theorem, first look at the constant term of the polynomial (call this number `p`):

`f(x)=3x^3-5x^2-12x+color(red)(20)`

Then find all the factors of that number:

`1, 20, 2, 10, 4, 5` and `-1, -20, -2, -10, -4, -5`

Next look at the leading coefficient (call this number `q`):

`f(x)=color(red)(3)x^3-5x^2-12x+20`

Again find the factors:

`1, 3` and `-1, -3`

Now list all the possible factors of `p` divided by all the factors of `q`:

`+-1, +-20, +-2, +-10, +-4, +-5, +-1/3, +-20/3, +-2/3, +-10/3, +-4/3, +-5/3`

Now you have to try plugging in all these numbers for `x` to find out which one makes `f(x)=0`

After a lot of computation, you will find that `-2, 2, and 5/3` make `f(x)=0`, so:

`f(x)=3x^3-5x^2-12x+20=(x+2)(x-2)(x-5/3)`