Question #66e6f

1 Answer
Nov 13, 2017

#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)#