Finding
Rational Zeros of Polynomials
Factor Theorem
If
is a zero of the polynomial ,
then is
a factor of .
Conversely, if
is a factor of .
then is a zero of .
By the Division Algorithm
.
By the Remainder Theorem
.
If
is a zero of ,
then ,
so .
In words this says that
is a factor of .
Conversely, if
is a factor of ,
we can write
.
Then
which says that
is a zero of .
See Example 1, pages 298 - 299.
The Fundamental Theorem of Algebra
Every polynomial of degree
has at least one zero.
This theorem implies that
Every polynomial of degree
has exactly
zeros.
Example
has exactly 2 zeros.
has exactly 17 zeros.
The zeros are not necessarily distinct.
has 2001 zeros.
The zero occurs
2001 times.
4 is said to be a root of multiplicity 2001.
Example
Factor
and solve the equation .
The first step is, essentially, trial and error as we search
for a linear factor of
of the form
.
Try .
Use synthetic division to see if .
first line
second line
third line is the quotient and the remainder is the
last number
,
not ,
so
is not a factor and
is not a solution of the equation.
Next, try
or
in the form ,
so .
Since ,
we know that
is one factor,
.
The coefficients of the quotient are given by the numbers on the third line,
except for the last number, ,
which is the remainder.
Since the dividend
is a third degree polynomial
and the divisor
is a first degree polynomial,
the quotient must be a second degree polynomial,
and the first three numbers on the third line
must be the coefficients of that second degree polynomial.
.
So
.
is easily factored as .
is the complete factoring.
To solve the equation ,
use the Zero Product Principle.
Set each factor equal to zero and solve the equations.
So the solutions are
NOTE: If the quadratic
is not easily factorable,
use the quadratic formula
to solve the equation .
is called the discriminant.
See Example 2, page 301.
Complex Zeros
In the quadratic formula, when the discriminant is less than zero
the roots of the quadratic equation are complex conjugate numbers.
Example
The roots of
where
are the complex conjugate numbers
.
Complex Zeros Theorem
Complex zeros, if they exist, of a polynomial with real coefficients
occur in conjugate pairs.