3.4 Theorems about Zeros of Polynomial Functions
The Fundamental Theorem of Algebra
The multiplicities of the complex (including the real) zeros of a polynomial add up to the degree of
the polynomial.
Any non-real zeros of a polynomial function with real coefficients must occur in conjugate pairs:
non-real zeros
real coefficients
complex
Certain irrational zeros of a polynomial function with rational coefficients also occur in conjugate
pairs:
irrational zeros
rational coefficients
real
Rational Zeros Theorem
If a rational number
in lowest terms is a zero of a polynomial function
with integer coefficients, then divides and divides
rational zeros
rational
.
integer coefficients
Descartes’ Rule of Signs
Suppose a polynomial function
term is , factor out a power of .)
has real coefficients and nonzero constant term. (If the constant
1. Write the terms of
in order of descending degree, leaving out terms with coefficient .
2. Let be the number of variations of sign in
.
3. Let be the number of variations of sign in
.
The number of positive real zeros of
The number of negative real zeros of
is one of the numbers
is one of the numbers
.
.
3.4 Examples
Find a polynomial function of degree 3 with the given numbers as zeros:
1.
2.
3.
(multiplicity 3), (multiplicity 1)
Find a polynomial function (in factored form) of lowest degree with rational coefficients which
has the given numbers as some of its zeros:
4.
5.
6.
Assuming that the polynomial function has the given zero, find the other zeros:
7.
8.
List all possible rational zeros of the function:
9.
10.
Factor the polynomial into linear factors:
11.
What does Descartes’ Rule of Signs tell you about the possible numbers of positive real,
negative real, and non-real zeros of the function?
12.
13.