In this lesson, you will use the rational zero theorem to determine the possible rational zeros of a polynomial function
which in turn will be used to find all zeros of
Specifically, this lesson will cover:
1. Using the Rational Zero Theorem to Find Rational Zeros
Just like we have seen with quadratic equations, polynomial equations can have several types of zeros: rational, irrational, pure imaginary, and/or nonreal complex.
In the last tutorial, we were able to solve a polynomial equation when given one of the polynomial’s factors. This isn’t always realistic, so we need a way to determine a set of possible zeros.
As a result of the following theorem, we will be able to determine which rational zeros are possible for a given polynomial
We should define the rational zero theorem. Given a polynomial 
where 
are all integers, then every rational zero of 
has the form 
where p is a factor of the constant term 
and q is a factor of the leading coefficient 
This is often written 
Note: 
could be positive or negative.
As a special case, when the leading coefficient is 1, then the possible rational zeros are the factors (positive or negative) of the constant term.
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EXAMPLE
Consider the function
Then, the possible rational zeros are
The factors of 6 are 1, 2, 3, and 6. The factors of 2 are 1 and 2. To find the possible rational zeros, divide each factor in the numerator by each factor in the denominator. There may be duplicates, so be sure not to list them twice in the final response.
Performing the divisions, we have
Since
and
, and they are already listed, omit those choices. Then, be sure to affix a “
” sign in front of each number.
Then, the possible rational zeros are
and
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In this video, we will list all the possible rational zeros of
-
Consider the function
There is no guarantee that any of the possibilities are actually zeros of the function—the theorem only provides possibilities.
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- Rational Zero Theorem
- Given a polynomial where are all integers, then every rational zero of has the form where p is a factor of the constant term and q is a factor of the leading coefficient
2. Finding the Zeros of Polynomial Functions
We now have a process to solve a polynomial equation.
-
- Make sure that all coefficients of the terms in the polynomials are integers.
- Use the rational zero theorem to write all possible rational zeros.
- Use synthetic division to determine which (if any) of the numbers in step 2 are zeros.
- Repeat step 3 until you arrive at a quadratic function as a quotient.
- Solve the remaining quadratic equation using any convenient method (the quadratic formula may be needed).
Note: remember that it is possible for a zero to occur more than once. This means that it is possible for a zero to have a multiplicity greater than one.
Also, here is a helpful hint.
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If the sum of the coefficients of a polynomial function is 0, then
is a zero of the polynomial function.
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EXAMPLE
Consider the function
Find all zeros of
First, list the possible rational zeros. Since the leading coefficient is 1, these are the factors of 6.
The possible rational zeros are
and
Next, since we do not know a factor of
we will use synthetic division to find a factor.
Note that the sum of the coefficients of
is not zero; therefore,
is not a zero.
Let’s try
Since the remainder is nonzero,
is not a zero of
Now let’s try
Success! This means
is a zero of
In factored form,
The remaining zeros can be found by solving
Since
doesn’t factor, the quadratic formula is used.
Then, the remaining zeros are
Note: these are not rational.
Thus,
has three distinct zeros: 2,
and
-
EXAMPLE
Consider the function
Find all zeros of
First, list the possible rational zeros:
Notice that the sum of the coefficients of
is 0. This means that
is a zero.
Next, perform the division:
In factored form,
The remaining zeros can be found by solving
or in factored form,
This means that
is a zero with multiplicity 2.
Thus, the zeros of
are 1 and
(multiplicity 2).
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Consider the function
Find all zeros of f (x ).First, find the possible rational zeros:
Strategy: Find one rational zero (hopefully), then perform division. At this point, we will be left with a quadratic that can either be factored or solved using the quadratic formula.
It turns out that -1 is a zero. Here is the table from synthetic division:
The factored form of
is
Next, the quadratic factors, the completely factored form of
is
We already know that
is a zero. To find the others, set each linear factor equal to 0 and solve.
The result: The zeros of
are
2, and
In this lesson, you learned how to use the rational zero theorem to find a list of possible rational zeros of a polynomial function, which then provides a process to find all the zeros of a polynomial function. After obtaining the possible zeros, trial and error is used to determine which are zeros, which enables us to write the polynomial in factored form. Once the quotient is quadratic, there is no need to perform trial and error anymore since quadratics can be solved with a variety of methods (usually, factoring or the quadratic formula are used). Keep in mind that it is possible that a polynomial has no rational zeros.
SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.
and 
and 
where 
are all integers, then every rational zero of 
has the form 
where p is a factor of the constant term 
and q is a factor of the leading coefficient 
