Difference between revisions of "Real Zeros of a Polynomial Function"

From Math Wiki
Jump to navigation Jump to search
(Created page with "<div class="noautonum">__TOC__</div> ==Introduction== This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that say...")
 
 
Line 15: Line 15:
 
The first theorem makes a statement about the number of zeros:
 
The first theorem makes a statement about the number of zeros:
  
−
'''Theorem'''A polynomial function of degree n cannot have more than n real zeros.
+
  '''Theorem''' A polynomial function of degree n cannot have more than n real zeros.
  
 
When a polynomial is written in standard form, decreasing degree order, we have even more information about the
 
When a polynomial is written in standard form, decreasing degree order, we have even more information about the
 
potential zeros of a polynomial.
 
potential zeros of a polynomial.
  
−
'''Theorem: Descartes' Rule of Signs'''
+
  '''Theorem: Descartes' Rule of Signs'''
−
Let f denote a polynomial function in standard form.
+
  Let f denote a polynomial function in standard form.
−
The number of real positive zeros of f is equal to the number of variations of sign of the nonzero coefficients of f(x)
+
  The number of real positive zeros of f is equal to the number of variations of sign of the nonzero coefficients of f(x) or else equals that number
−
or else equals that number minus an even integer.
+
  minus an even integer.
 
   
 
   
−
The number of negative zeros of f either equals the number of variations in the sign of the nonzero coefficients
+
The number of negative zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(-x) or else equals that number minus an even integer.
−
of f(-x) pr else equals that number minus an even integer.
 
 
   
 
   
 
   
 
   
−
Example:
+
Example:
−
Let <math> f(x) = x^3 - 4x^2 - 6x + 9</math>
+
Let <math> f(x) = x^3 - 4x^2 - 6x + 9</math>
−
+
 
−
To count the number of positive roots, we note that there are 2 sign changes, from 1 to -4 and -6 to 9.
+
To count the number of positive roots, we note that there are 2 sign changes, from 1 to -4 and -6 to 9.
−
So there either 2 or 0 positive roots.
+
So there either 2 or 0 positive roots.
 
   
 
   
−
To count the number of negative roots, we have to look at  
+
To count the number of negative roots, we have to look at  
−
<math>f(-x) = -x^3 -4x^2 +6x + 9</math>. Here we only have one sign change, from -4 to 6. So there is one negative root.
+
<math>f(-x) = -x^3 -4x^2 +6x + 9</math>. Here we only have one sign change, from -4 to 6. So there is one negative root.
 
   
 
   
−
Finally we arrive at the most precise theorem for finding rational zeros.
+
Finally we arrive at the most precise theorem for finding rational zeros.
 
   
 
   
−
'''Rational Zeros Theorem'''
+
  '''Rational Zeros Theorem'''
−
Let f be a polynomial function of degree 1 or higher of the form <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots
+
  Let f be a polynomial function of degree 1 or higher of the form <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots
 
   + a_1x + a_0 ~ a_n \neq 0 ~ a_0 \neq 0</math>
 
   + a_1x + a_0 ~ a_n \neq 0 ~ a_0 \neq 0</math>
 
   where each coefficient is an integer. If <math> \frac{p}{q}</math>, in lowest terms, is a rational zero of f, then p must divide <math>a_0</math>,
 
   where each coefficient is an integer. If <math> \frac{p}{q}</math>, in lowest terms, is a rational zero of f, then p must divide <math>a_0</math>,
 
   and q must divide <math>a_n</math>
 
   and q must divide <math>a_n</math>
 
    
 
    
−
  [[Math_5|'''Return to Topics Page]]
+
[[Math_5|'''Return to Topics Page]]

Latest revision as of 20:07, 20 October 2015

Introduction

This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that says if f(x) is a polynomial and f(r) = 0 for a real number r, then x - r is a factor of f(x).

Factor Theorem

The mentioned important fact is restated here as teh factor theorem:

Let f be a polynomial function. Then x - c is a factor of f(x) if and only if f(c) = 0.

Theorems For Finding Zeros

The first theorem makes a statement about the number of zeros: 

 Theorem A polynomial function of degree n cannot have more than n real zeros.

When a polynomial is written in standard form, decreasing degree order, we have even more information about the potential zeros of a polynomial. 

 Theorem: Descartes' Rule of Signs
 Let f denote a polynomial function in standard form.
 The number of real positive zeros of f is equal to the number of variations of sign of the nonzero coefficients of f(x) or else equals that number
 minus an even integer.

The number of negative zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(-x) or else equals that number minus an even integer.


Example: Let

To count the number of positive roots, we note that there are 2 sign changes, from 1 to -4 and -6 to 9. So there either 2 or 0 positive roots.

To count the number of negative roots, we have to look at . Here we only have one sign change, from -4 to 6. So there is one negative root.

Finally we arrive at the most precise theorem for finding rational zeros. 

 Rational Zeros Theorem
 Let f be a polynomial function of degree 1 or higher of the form 
 where each coefficient is an integer. If , in lowest terms, is a rational zero of f, then p must divide ,
 and q must divide

Return to Topics Page

Retrieved from "https://wiki.math.ucr.edu/index.php?title=Real_Zeros_of_a_Polynomial_Function&oldid=1122"