College Algebra, 2013a

270 Polynomial Functions
Multiplying both sides of this equation by q n , we clear the denominators to get
Rearranging this equation, we get
a n p n + a n−1 p n−1 q + ...+ a 1 pq n−1 + a 0 q n =0
a n p n = −a n−1 p n−1 q − ...− a 1 pq n−1 − a 0 q n
Now, the left hand side is an integer multiple of p, and the right hand side is an integer multiple of
q. (Can you see why?) This means a n p n is both a multiple of p and a multiple of q. Sincep and q
have no common factors, a n must be a multiple of q. If we rearrange the equation
a n p n + a n−1 p n−1 q + ...+ a 1 pq n−1 + a 0 q n =0
as
a 0 q n = −a n p n − a n−1 p n−1 q − ...− a 1 pq n−1
we can play the same game and conclude a 0 is a multiple of p, and we have the result.
Example 3.3.2. Let f(x) =2x 4 +4x 3 − x 2 − 6x − 3. Use the Rational Zeros Theorem to list all
of the possible rational zeros of f.
Solution. To generate a complete list of rational zeros, we need to take each of the factors of
constant term, a 0 = −3, and divide them by each of the factors of the leading coefficient a 4 =2.
The factors of −3 are± 1and± 3. Since the Rational Zeros Theorem tacks on a ± anyway, for
the moment, we consider only the positive factors 1 and 3. The factors of 2 are 1 and 2, so the
Rational Zeros Theorem gives the list { ± 1 1 , ± 1 2 , ± 3 1 , ± 3 {
2}
or ±
1
2 , ± 1, ± 3 2 , ± 3} .
Our discussion now diverges between those who wish to use technology and those who do not.
3.3.1 For Those Wishing to use a Graphing Calculator
At this stage, we know not only the interval in which all of the zeros of f(x) =2x 4 +4x 3 −x 2 −6x−3
are located, but we also know some potential candidates. We can now use our calculator to help
us determine all of the real zeros of f, as illustrated in the next example.
Example 3.3.3. Let f(x) =2x 4 +4x 3 − x 2 − 6x − 3.
1. Graph y = f(x) on the calculator using the interval obtained in Example 3.3.1 as a guide.
2. Use the graph to shorten the list of possible rational zeros obtained in Example 3.3.2.
3. Use synthetic division to find the real zeros of f, and state their multiplicities.
Solution.
1. In Example 3.3.1, we determined all of the real zeros of f lie in the interval [−4, 4]. We set
our window accordingly and get