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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, p(x) ,

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


Synthetic Division

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

Divide 2x45x314x237x30 by x2 .


Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
  1. (x3+6x2+7x+10)÷(x+2)
  2. (2x415x330x220x+42)÷(x+9)
  3. (3x5+4x3x2)÷(x1)
  4. Find f(2) if f(x)=2x45x310x2+21x4
Find all real zeros of the following polynomials, given one or two zeros.
  1. x35x22x+10;2
  2. x4+7x3+6x232x32;4,1
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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  5x53x4+2x3+x27x+3

  2. Use Descartes Rule of Signs to identify the possible number of positive and negative roots of f(x)=3x37x2+8x2

  3. Find the root(s) of f(x)=2x35x24x+3

  4. Graph the polynomial function  x35x22x+10  by using synthetic division to find the x intercepts and locate the y 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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