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.

Example
If    is a root then    also is a root, and vise versa.

If a polynomial with real coefficients is of odd degree
- for example,      -
it could have three real zeros.
However, if it has one complex zero then automatically it has another
complex zero, namely, the complex conjugate.
That make two zeros. We know the polynomial must have exactly three zeros.
So the remaining zero must be real, otherwise, there would be two more
complex roots and the total number of zeros would be four, which is impossible.
This result is the

Odd-Degree Polynomial Real Zero Theorem
A polynomial of odd degree with real coefficients has at least one real zero.


Rational Zeros Theorem
If the polynomial with integer coefficients
   
has a rational zero, reduced to lowest terms,
                                 ,
then    must be a factor of    and    must be a factor of   .


Note that these theorem only says that if a polynomial has a rational zero
it must be on the list of possible rational zeros given by the theorem.
If we check each number on the list and none of then are zeros,
that only means that the polynomial has no rational roots.
The zeros must then be either irrational numbers or complex numbers.

Example
        has no rational roots.
                              Its roots are the irrational numbers   .
      has no rational roots.
                              Its roots are the complex conjugate numbers    .

See Examples 3 - 5, pages 304 -307.


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