Example: prove that the roots of the below given polynomial satisfy the rational root theorem.

P ( x ) = 12x³ - 41x² - 38x + 40 = ( x - 4) (3x - 2) ( 4x +5)

Solution

From the factored form we can illustrates that the zeroes are,

                           x= 4 = 4/1         x= 2/3                x= - 5/4

Notice that we wrote the integer like a fraction to fit it into the theorem.  Also, along with the negative zero we can put the -ve onto the numerator or denominator.  It won't issue.

Thus, in according to the rational root theorem the numerators of these fractions (with or without the minus sign on the third zero) have to all be factors of 40 and the denominators have to all be factors of 12.

Here are various ways to factor 40 & 12.

40 =( 4) (10)               40 = ( 2) ( 20)      40 = (5) (8)         40 = ( -5) ( -8)

12 = (1) (12)              12 = (3) ( 4)                                12 = ( -3) ( -4)

From these we can illustrate that actually the numerators are all factors of 40 and the denominators are all factors of 12.  Also notice that, as illustrated, we can put the minus sign on the third zero on either the numerator or the denominator and still it will be a factor of the suitable number.