Section 2.5: Zeros of Polynomial Functions (Day 1)
These two theorems together mean that when working with complex numbers (a + bi)

Every nth degree polynomial has exactly n zeros.
AND

Each of the zeros of a polynomial has a factor.
These are existence theorems. They don’t help us find the zeros. They just tell us they exist.
Examples of zeros of polynomial functions that illustrate these two theorems:
Now, let’s assume you don’t have a graphing calculator and unlike on the last test we took, you aren’t
told any of the possible zeros of a function. How can you find the zeros without a graph and without a
place to start?
For example, what if I gave you the following polynomial and asked you to find the zeros?
How would we know where to start without a graph?
We could use synthetic division, but what numbers would we test?
This is where the rational zero test can be helpful. It gives us a list of possible rational numbers that
could be zeros of the function. This gives us a place to start our test instead of completely flying blind.
Before you look at the rational zero test, remember a rational number is any number that can be written
as a fraction. So rational numbers are the integers, fractions, and any decimals that end or repeat.
Notice, this theorem only works for
polynomials with integer coefficients.
To use this theorem to make a list of possible rational zeros



make a list of all the factors of the constant term (p)
make a list of all the factors of the leading coefficient (q)
divide each factor of p by each factor of q
This list is now used to give you a starting point to start testing zeros using synthetic division. Without
this list and without a graphing calculator it would be nearly impossible to find all the zeros of a
polynomial with a large degree.
Example: Let’s change the directions circled and say we want to find all the zeros regardless of whether
they are rational, irrational, or complex.
The factors of the constant (6)
are 1, 2, 3, 6
The ONLY factor of the leading
coefficient (1) is 1
If we divide every factor of p by
the only factor of q(1) we just get
the list of factors of p
And further, if we solve each factor, the zeros of f are x  1, 2,  i 3
The degree of the polynomial was 4 and we have 4 zeros.
All the testing is not shown here.
They may have tested other
numbers using synthetic division
before they found the x values
that would give a remainder of
zero. Maybe they tested 1,
found it wasn’t a zero, then
moved on to -1. Also notice they
used the answer from the first
synthetic division to perform the
next synthetic division.
Another example
Once again, all the testing is not
shown here. They may have
tested other numbers using
synthetic division before they
found the x value that would
give a remainder of zero.
Assignment: p.160 #11-31 odds, 37-41 odd (14 questions)
Change of directions on this assignment:
When you are asked to find the zeros show ALL of these steps
1. List all possible rational zeros using the rational zero test (p/q)
2. Test the number from this list until you find those that give a remainder of zero.
3. Once you have the polynomial factored down to a quadratic, you can solve the quadratic using
factoring or quadratic formula
4. IMPORTANT: Assume you DON’T have a graphing calculator. On your next test you won’t be
able to use one on at least part of the test!
For #11-23, find ALL the zeros (this means not just real zeros, but also any complex zeros)
On #25 and #27 DO NOT sketch the graph and DO NOT use your graphing calculator to help find the
zeros.
On #29 and #31 you CAN use your graphing calculator to help find the zeros, but you have to test using
synthetic division to find the zeros! (The calculator is used to guide you, it doesn’t replace your work.)
Hint for #37-41: Write each zero as a factor and multiply them out to find the polynomial that would
give those zeros.
p.160 #11-31 odds, 37-41 odd (14 questions) with KEY
KEY