1
Section 4.5
The Zeros of Polynomial Functions; The Fundamental Theorem of Algebra
The goal of this section is to be able to find the zeros of polynomial functions or equivalently, we are
interested in solving polynomial equations of the form f ( x)  0 . To begin our discussion we must take a
look at a very important theorem that will help us to create a list of potential zeros of a polynomial function.
Objective 1
Using the Rational Zeros Theorem
We are able to find all real zeros of f ( x)  x3  6 x  4 given that x  2 was a zero. Because we knew that
x  2 was a zero, we were able to use the Factor Theorem and synthetic division to rewrite f as
f ( x)  ( x  2)( x 2  2 x  2) . We then solved the quadratic equation x2  2x  2  0 to find the other two
zeros.
But what if we were not given the fact that x  2 was a zero? How do we start looking for the zeros? Is
there a systematic way to determine possible zeros of a polynomial function? The answer to this question is
yes if we are given a polynomial with integer coefficients. If a polynomial has integer coefficients, then we
are able to create a list of the potential rational zeros using The Rational Zeros Theorem.
The Rational Zeros Theorem
Let f be a polynomial function of the form f ( x)  an xn  an1xn1  an2 xn2 
p
 a1x  a0 of degree n  1
p
where each coefficient is an integer. If q is a rational zero of f (where q is written in lowest terms), then
p must be a factor of the constant coefficient, a0 , and q must be a factor of the leading coefficient an .
The Rational Zeros Theorem can only provide us with a list of possible rational zeros. It does not
guarantee that the polynomial will have a zero from the list. Simply stated, if a polynomial with
integer coefficients has a rational zero, then it must be on the list created using this theorem.
2
Objective 3 Finding the Zeros of a Polynomial Function
Before we begin to find the zeros of polynomial functions, we should know how many zeros to expect. The
Fundamental Theorem of Algebra says that every polynomial of degree n  1 has at least one
complex zero.
The Fundamental Theorem of Algebra
Every polynomial function of degree n  1has at least one complex zero.
Suppose f ( x) is a polynomial function of degree n  1. By the Fundamental Theorem of Algebra, f has at
least one complex zero, call it c1 . By the Factor Theorem, x  c1 is a factor of f ( x) and it follows that we
can rewrite f ( x) as f ( x)  ( x  c1 )q1 ( x) where q1 ( x) is another polynomial. If q1 ( x) is of degree 1 or more
then we repeat the process again and rewrite f ( x) as p( x)  ( x  c1 )( x  c2 )q2 ( x) where c2 is a zero of q1 ( x) .
If f ( x) is a degree n polynomial then we can repeat this process a total of n times to rewrite f ( x) as
f ( x)  a( x  c1 )( x  c2 )( x  c3 )  x  cn  where c1 , c2 , c3 cn are zeros of f ( x) and a is the leading
coefficient.
Therefore, every polynomial of degree n has n complex zeros and can be written in completely factored
form. Note that some of the zeros could be the same and need to be counted each time as indicated by the
following theorem.
The Number of Zeros Theorem
Every polynomial of degree n has n complex zeros provided each zero of multiplicity
greater than one is counted accordingly.
Note: Once we factor a polynomial into the product of linear factors and a quadratic function of the form
f ( x)  ( x  c1 )( x  c2 )( x  c3 )  x  ck  ax 2  bx  c , we need simply to solve the quadratic equation


ax 2  bx  c  0 to find the remaining two zeros. For example, if we want to find all zeros of
g ( x)  2 x3  3x 2  4 x  3 we need only try to find one zero from the list created by the Rational Zeros
Theorem then find the two zeros of the remaining quadratic function. See if you can find the zeros of
g ( x)  2 x3  3x 2  4 x  3 . Watch this video to see if you are correct!
3
Objective 4
Solving Polynomial Equations
Solving polynomial equations of the form f ( x)  0 is equivalent to finding the zeros of f ( x) . For example,
the solutions to the equation 6 x 4  13x3  61x 2  8 x  10  0 are exactly the zeros of the polynomial
1
1
function f ( x)  6 x4  13x3  61x 2  8x  10 . The real solutions of this equation are  and while the two
2
3
non-real solutions are 1  3i and 1  3i .
4
Objective 7
Sketching the Graphs of Polynomial Functions
Now that we have a procedure for finding the zeros of a polynomial function we can modify the four-step
process that was discussed in section 4.3 and put it all together to sketch the graphs of polynomial functions
of the form f ( x)  an xn  an1xn1  an2 xn2   a1x  a0 .
Steps for Sketching the Graphs of Polynomial Functions
1.
2.
3.
4.
Determine the end behavior.
Plot the y-intercept f (0)  a0 .
Use the Rational Zeros Theorem, the Factor Theorem and synthetic division, or
the Intermediate Value Theorem to find all zeros and completely factor f.
Choose a test value between each real zero and complete the graph.
(Remember, without calculus, there is no way to precisely determine the coordinates of the
turning points)