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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)\begin{align*}p(x)\end{align*} ,

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

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### Synthetic Division

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

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











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##### 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*}
2. \begin{align*}(2x^4-15x^3-30x^2-20x+42) \div (x+9)\end{align*}
3. \begin{align*}(3x^5+4x^3-x-2) \div (x-1)\end{align*}
4. Find \begin{align*}f(-2)\end{align*} if \begin{align*}f(x)=2x^4-5x^3-10x^2+21x-4\end{align*}
##### 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*}
2. \begin{align*}x^4+7x^3+6x^2-32x-32; -4, -1\end{align*}
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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*}

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*}

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

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