Using the Rational Zeros Theorem
Trigonometry and Advanced Math - Section 6.6
Warm-Up Problem
Given f ( x)  2 x 3  7 x 2  33x  18 and that f(-6) = 0.
a.
Use synthetic division to factor f(x) completely.
b.
What are the zeroes of f(x)?
c.
What is the prime factorization of the constant term, -18? List some factors of the constant term.
d.
What is the prime factorization of the leading coefficient, 2? List some factors of the leading coefficient.
Now consider g ( x)  64x 3  120x 2  34x 105 . The zeros of this function are 
3 5 7
, , .
2 4 8
e.
What is the prime factorization of the constant term, -105? List some factors of the constant term.
f.
What is the prime factorization of the leading coefficient, 64? List some factors of the 64.
g.
How do the factors of the leading coefficient and the constant term relate to the zeroes of a polynomial? Make a
conjecture.
This result generalizes to the following theorem:
The Rational Zero Theorem
Let f ( x)  a n x n  ... a 2 x 2  a1 x1  a0 x 0 such that all coefficients ai  . Every rational zero of f has the following form:
some factor of constant term, a 0
p

q some factor of leading coefficien t, a n
We know a few methods for finding the zeroes of quadratics, but when we get into higher-degree polynomials, things get a
bit sketchy, which is to say, difficult. One way that always gives us a good idea of the zeroes is to graph the polynomials, but
if we wish to be precise, we need other ways to search for zeros. We can use the rational zero theorem to help narrow down
this search.
How to use the rational zero theorem to factor or find the rational zeroes of a polynomial:
1.
Factor the constant term and leading coefficient. Use this information and the rational zeros theorem to LIST the
possible rational zeroes.
2.
CHOOSE which rational zeroes to check. If there are only a few, there's no harm in trying them all, but when you
have several possibilities, you don't want to try them all. (Remember that idea that mathematicians are lazy.) So, it's
usually a good idea to sketch the graph the polynomial to narrow down your search.
3.
TEST the values you've chosen. Use synthetic division. Remember that the last term under the bar in synthetic
division is the remainder. You're looking for remainders of zero. Why?
4.
Use the results of synthetic division to FACTOR out the binomial that corresponds to the zero you just found. In
factoring, rewrite the binomial with integer coefficients.
5.
If you can use another method (simple factoring, quadratic formula, factoring patterns like differences of squares,
Pascal's triangle, or factoring by group) to break down the remaining factor of the polynomial, do so. Otherwise,
repeat this process for the remaining factor.
Be careful to answer the question you're asked -- if you're asked to factor a polynomial, you'll give a factored form of a
polynomial as a result. If you're asked for zeroes, you need to use the zero product property to find the zeroes.
Let's look at some examples.
Example 1: Find the rational zeroes of f ( x)  x 3  2 x 2  11x  12 .
Example 2: Find all real zeroes of f ( x)  10x 4  3x 3  29x 2  5x  12 .
Modeling Problem:
You are designing a candle-making kit. Each kit will contain
25 cubic inches of candle wax and a mold for making a model
of the pyramid-shaped building at the Louvre Museum in
Paris1. You want the height of the candle to be 2 inches less
than the length of each side of the candle's square base. What
should the dimensions of your candle mold be?
Recall: The volume of a pyramid is given by V 
1
Bh ,
3
where B is the area of the base and h is the height.
Preliminary Questions:
What is the volume? ______________
What are the units? _______________
What is the length of a side of the base? _______________
What are the units? _______________
What is the area of the base? _________________
What are the units? _______________
What is the height? ________________
What are the units? _______________
Model and solution:
Now set up a model and solve the problem.
Final result: ________________________________________________________________________________
Homework: 6.6: #15-35 every other odd
1
Image taken from http://drivel.ca/europe/paris/louvre-pyramid1.jpg