Unit 4 – Polynomial/Rational Functions
Zeros of Polynomial Functions
(Unit 4.3)
William (Bill) Finch
Mathematics Department
Denton High School
Introduction
Theorems
Zeros
Complex Zeros
Summary
Lesson Goals
When you have completed this lesson you will:
W. Finch
Zeros
I
Find complex solutions to quadratic functions.
I
Apply the Fundamental Theorem of Algebra.
I
Find all complex zeros (real and imaginary) of polynomial
functions.
DHS Math Dept
2 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra
If f (x) is a polynomial of degree n, where n > 0, then f has at
least one zero in the complex number system.
Linear Factorization Theorem
If f (x) is a polynomial of degree n, where n > 0, then f has
precisely n linear factors
f (x) = an (x − c1 )(x − c2 ) · · · (x − cn )
where c1 , c2 , . . . , cn are complex numbers.
W. Finch
DHS Math Dept
Zeros
3 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Zeros of a Polynomial Function
Function
Degree
Linear Factors
Zeros
f (x) = x − 4
1
x −4
4
g (x) = x 2 − 4x + 4
2
(x − 2)(x − 2)
2, 2
h(x) = x 3 + 9x
3
x(x ± 3i)
0, ±3i
4
(x ± 2)(x ± 2i)
±2, ±2i
4
j(x) = x − 16
Degree = # of Linear Factors = # of Zeros
W. Finch
Zeros
DHS Math Dept
4 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Finding the Zeros of a Polynomial Function
To find the zeros of a polynomial function you may use one of
the following tools (or a combination of them):
I
Graphing
I
Factoring
I
Division (long or synthetic)
In order to get started it is often helpful to identify a
rational zero by either graphing or with the
Rational Zero Test.
W. Finch
DHS Math Dept
Zeros
5 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Identifying Rational Zeros
The Rational Zero Theorem
If the polynomial f (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 has
integer coefficients, every rational zero of f has the form
Rational zero =
p
q
where p and q have no common factors other than ±1, and
p = an integer factor of the constant term a0
q = an integer factor of the leading coefficient an
W. Finch
Zeros
DHS Math Dept
6 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Example 1
For the polynomial f (x) = x 3 − 15x 2 + 75x − 125
a) List all possible rational zeros.
b) Determine which are actually zeros of f .
W. Finch
DHS Math Dept
Zeros
7 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Example 2
For the polynomial g (x) = 2x 4 − 9x 3 − 18x 2 + 71x − 30
a) List all possible rational zeros.
b) Determine which are actually zeros of g .
W. Finch
Zeros
DHS Math Dept
8 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Complex Zeros
Complex Zeros Occur in Conjugate Pairs
Let f (x) be a polynomial function that has real coefficients. If
a + bi (where b 6= 0) is a zero of the function, the conjugate
a − bi is also a zero of the function.
W. Finch
DHS Math Dept
Zeros
9 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Example 3
Find all of the zeros of f (x) = x 3 − 4x 2 + 21x − 34 if you
know one of the zeros is 1 + 4i.
W. Finch
Zeros
DHS Math Dept
10 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Example 4
For the polynomial h(x) = 8x 3 − 4x 2 + 6x − 3
a) Find all of the zeros of the function.
b) Find all of the linear factors of the function.
W. Finch
DHS Math Dept
Zeros
11 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Example 5
Write a third-degree polynomial function whose zeros include
2 and 7i.
W. Finch
Zeros
DHS Math Dept
12 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
Example 6
The water level in a bucket sitting on a patio can be modeled
by
f (x) = x 3 + 4x 2 − 2x + 7
where f (x) is the height of the water in millimeters and x is
the time in days. On what day(s) will the water reach a height
of 10 millimeters?
W. Finch
DHS Math Dept
Zeros
13 / 14
Introduction
Theorems
Zeros
Complex Zeros
Summary
What You Learned
You can now:
W. Finch
Zeros
I
Find complex solutions to quadratic functions.
I
Apply the Fundamental Theorem of Algebra.
I
Find all complex zeros (real and imaginary) of polynomial
functions.
I
Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39,
49-53 odd, 61-65 odd
DHS Math Dept
14 / 14