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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

Vocabulary

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

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

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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 .

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Synthetic Division

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

Divide 2x^4-5x^3-14x^2-37x-30 by x - 2 .




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Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
  1. (x^3+6x^2+7x+10) \div (x+2)
  2. (2x^4-15x^3-30x^2-20x+42) \div (x+9)
  3. (3x^5+4x^3-x-2) \div (x-1)
  4. Find f(-2) if f(x)=2x^4-5x^3-10x^2+21x-4
Find all real zeros of the following polynomials, given one or two zeros.
  1. x^3-5x^2-2x+10; -2
  2. x^4+7x^3+6x^2-32x-32; -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  5x^{5}-3x^{4}+2x^{3}+x^{2}-7x+3

  2. Use Descartes Rule of Signs to identify the possible number of positive and negative roots of f(x)=3x^{3}-7x^{2}+8x-2

  3. Find the root(s) of f(x)=2x^{3}-5x^{2}-4x+3

  4. Graph the polynomial function  x^3-5x^2-2x+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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