Verify that the roots of the following polynomial satisfy the rational root theorem.

Solution:

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

Notice that we wrote the integer as a fraction to fit it into the theorem. Also, with the negative zero we can put the negative onto the numerator or denominator.

According to the rational root theorem the numerators of these fractions (with or without the minus sign on the third zero) must all be factors of 40 and the denominators must all be factors of 12.

Here are several ways to factor 40 and 12.

From these we can see that in fact the numerators are all factors of 40 and the denominators are all factors of 12. Also note that, as shown, we can put the minus sign on the third zero on either the numerator or the denominator and it will still be a factor of the appropriate number.

ZEROES OF POLYNOMIAL FUNCTION

Process for Finding Rational Zeroes

Use the rational root theorem to list all possible rational zeroes of the polynomial

Evaluate the polynomial at the numbers from the first step until we find a zero. Let’s suppose the zero is

then we will know that it’s a zero because

Once this has been determined that it is in fact a zero write the original polynomial as

Repeat the process using

this time instead of

This repeating will continue until we reach a second degree polynomial. At this point we can solve this directly for the remaining zeroes.