Approximating the Real Zeros of Polynomials

We will consider only the real zeros of polynomials with real coefficients.

Location Theorem
If    is a continuous function on an interval   ,
  and    are two numbers in   , and   and 
are of opposite sign, then    has at least one x-intercept
between    and   .

Geometrically, this is obvious.
Remember that a continuous graph is drawn without lifting the pencil
off the sheet of paper. The graph is unbroken, is in one solid piece,
there are no breaks or jumps in the graph.
So if the graph has a point below the x-axis and another point above the
x-axis, the graph, in order to get from the first point to the second point,
must cross the x-axis somewhere between the two points.
In other words, it has an x-intercept between the two points.

                                                                                              (b, f(b))
       
                                      (a, f(a))

See Figure 2, page 311.

This theorem can be used to locate the real zeros of a function.

See Example 1, page 312.

Any number that is larger than or equal to the largest zero of a polynomial
is called an upper bound of the zeros of the polynomial.
Any number that is less than or equal to the smallest zero of a polynomial
is called a lower bound of the zeros of the polynomial.

Using synthetic division, the following theorem
will enable us to determine a region within which all the real zeros
of a polynomial must be found.

Given an nth-degree polynomial   ,   ,   , and    
divided by    using synthetic division:
Upper Bound of Real Zeros
  If  and all the numbers in the quotient line of the synthetic division,
   including the remainder, are nonnegative, then is an upper bound of the
   real zeros of .

Lower Bound of Real Zeros
  If    and all the numbers in the quotient line of the synthetic division,
  including the remainder, alternate in sign, then    is a lower bound
  of the real zeros of   .

See Example 2, pages 313 - 314.

The above process locates where the real zeros are.
The following method is a way of actually approximating
the value of a real zero.

Again, is a polynomial with real coefficients.
The Bisection Method
If has opposite signs at the endpoints of the interval ,
then a real zero lies in the interval.
Bisect the interval. The midpoint is .
Check the sign of .
Choose the interval or the interval on which
has opposite signs at the endpoints.
Repeat this bisecting process until we get the number of decimal places
we want for the approximation of the zero. At any point in the process
if   , we stop, since    is a real zero.

See Example 3 - 6, pages 314 - 319.


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