Polynomial Functions and Graphs

Definitions
Given the function ,
the number    is a zero of the function if  .
Given the equation ,
the number    is a root or a solution of the equation if  .
Notice that asking for the x-intercept of the graph of 
is asking for the solutions of the equation .
The zero, root, solution, x-intercept produce the same number for the answer.

     
where the coefficients may be complex numbers,
is the general nth-degree polynomial function.


Synthetic Division
is a streamlined way of doing algebraic long division ONLY
when the divisor is of the form   .

For algebraic long division, see Example 1, page 286 - 287.

The following example illustrates the general process.
Divide    by synthetic Division.
                         dividend              divisor

Step 1)
Write the dividend and the divisor in descending powers of the variable,
 in this example.

Step 2)
If there are any missing powers of   in the dividend or in the divisor,
rewrite the polynomials showing the missing powers with zero coefficients.
In the example,    and  are missing in the dividend, so we rewrite
     .

Step 3)
Write the coefficients only.
The divisor must be written in form  ,
     
                                   is the number to put in the corner box..


          first line
                          second line
           _________________________
                      third line

Step 3) The Process.
Bring down the  .
Multiply it by what's in the corner box,   , and put the product,   ,
on the second line as shown above.
Add the product,   , to the corresponding number,   , on the first line,
put the answer,   , on the third line.
Note that we add, not subtract.

Repeat this process. We should get the numbers shown in the example.

Step 4) The Answer.
The numbers on the third line are the coefficients of the quotient, except for
the last number,   , which is the remainder in the algebraic long division.
Whatever the degree of the polynomial in the dividend, the degree of the
polynomial in the quotient is one less.

In the example, the dividend is a polynomial of the 3^rd degree, so the quotient
is a polynomial of the 2^nd degree.

The full answer to the long division problem is
      .

See Example 2, page 288 - 289.


The algebraic long division process may be written as

     
                     
                     
                     
                 _____
                           The remainder is a constant. It does not have the variable    in it.

Another way of writing the above, as a multiplication, is

              The Division Algorithm

This equation is an identity.
It is true for all values of   . So, in particular, it is true for  .

     
               

This result is called

The Remainder Theorem
Dividing the polynomial    by   , the remainder   
is the value of the polynomial    at  .
That is,
                 .

Let us check this for the above example,
         .
     
                   
This checks,
                        .

Synthetic division can be used to evaluate polynomials.
This example also shows that we found    by synthetic division.
is the last number on the third line. It is the remainder term.

See Example 3, page 290 - 291.


Graphing Polynomial Functions. The General Behavior.
Remember that points on the graph of a function  
can be written as   or, equivalently, as
     .
The second coordinate is just the value of the function   .

Let
     .

The leading term    dominates the behavior of the function.

     
     
                     

         left                            right
          
         Left and right both shoot up.




     
     
                         
                               right
          
                     left

     Left shoots down and right shoots up


     
      
                       

           
          left                               right

     Left and right both shoot down.



     
    
                
                      left
          
                                          right

      Left shoots up and right shoots down.


A turning point of a continuous graph is a point that separates
an increasing portion of the graph from a decreasing portion of the graph.
In the above examples the origin is a turning point for all the graphs.

Properties of Graphs of a Polynomial Function of degree
1) smooth graph, no breaks, no jumps
2) graph has at most     x-intercepts
3) graph has at most    turning points

Example

     


          


See Example 4,  pages 294 -295.


                        top
                        next     Finding Rational Zeros of Polynomials