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Real Zeros of Polynomials

Remainder, Factor, and Rational Zero Theorems and Descartes' Rule of Signs.

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Synthetic Division and Real Zeros of Polynomials


Complete the Theorem chart.
Name Theorem
Remainder Theorem ________________________________________________________________
Factor Theorem ________________________________________________________________
Rational Zero Theorem ________________________________________________________________


What form does the divisor need to be in to use synthetic division? ____________________


Fill in the blanks about Descartes Rule of Signs:

Given any polynomial,  ,

  1. Write it with the terms in ____________ order, i.e. from the ____________ degree term to the ____________ degree term.
  2. Count the number of ____________ of the terms in  . Call this .
  3. Then the number of ____________ of  is less than or equal to  .
  4. Further, the possible number of ____________ is ____________
  5. To find the number of ____________ of  , write  in descending order as above (i.e. change the sign of all terms in  with odd powers), and repeat the process above. Then the ____________ number of negative roots is  .


Synthetic Division

Here are the steps (via example) to synthetic division: 

Divide  by  .


Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
  1. Find  if 
Find all real zeros of the following polynomials, given one or two zeros.
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Real Zeros

  1. Use the rational zero theorem and synthetic division to find all the possible rational zeros of the polynomial  

  2. Use Descartes Rule of Signs to identify the possible number of positive and negative roots of 

  3. Find the root(s) of 

  4. Graph the polynomial function    by using synthetic division to find the  intercepts and locate the  intercepts.

  5. Write a 5th degree equation of a polynomial function with the zeroes: 0 (multiplicity 2), 2 (multiplicity 3), and -5 (multiplicity 2)

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