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


Synthetic Division

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

Divide \begin{align*}2x^4-5x^3-14x^2-37x-30\end{align*}2x45x314x237x30 by \begin{align*}x - 2\end{align*}x2 .


Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
  1. \begin{align*}(x^3+6x^2+7x+10) \div (x+2)\end{align*}(x3+6x2+7x+10)÷(x+2)
  2. \begin{align*}(2x^4-15x^3-30x^2-20x+42) \div (x+9)\end{align*}(2x415x330x220x+42)÷(x+9)
  3. \begin{align*}(3x^5+4x^3-x-2) \div (x-1)\end{align*}(3x5+4x3x2)÷(x1)
  4. Find \begin{align*}f(-2)\end{align*}f(2) if \begin{align*}f(x)=2x^4-5x^3-10x^2+21x-4\end{align*}f(x)=2x45x310x2+21x4
Find all real zeros of the following polynomials, given one or two zeros.
  1. \begin{align*}x^3-5x^2-2x+10; -2\end{align*}x35x22x+10;2
  2. \begin{align*}x^4+7x^3+6x^2-32x-32; -4, -1\end{align*}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  \begin{align*}5x^{5}-3x^{4}+2x^{3}+x^{2}-7x+3\end{align*}5x53x4+2x3+x27x+3

  2. Use Descartes Rule of Signs to identify the possible number of positive and negative roots of \begin{align*}f(x)=3x^{3}-7x^{2}+8x-2\end{align*}f(x)=3x37x2+8x2

  3. Find the root(s) of \begin{align*}f(x)=2x^{3}-5x^{2}-4x+3\end{align*}f(x)=2x35x24x+3

  4. Graph the polynomial function  \begin{align*}x^3-5x^2-2x+10\end{align*}x35x22x+10  by using synthetic division to find the \begin{align*}x-\end{align*}x intercepts and locate the \begin{align*}y-\end{align*}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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