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

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### Synthetic Division

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

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











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##### Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
1. (x3+6x2+7x+10)÷(x+2)\begin{align*}(x^3+6x^2+7x+10) \div (x+2)\end{align*}
2. (2x415x330x220x+42)÷(x+9)\begin{align*}(2x^4-15x^3-30x^2-20x+42) \div (x+9)\end{align*}
3. (3x5+4x3x2)÷(x1)\begin{align*}(3x^5+4x^3-x-2) \div (x-1)\end{align*}
4. Find f(2)\begin{align*}f(-2)\end{align*} if f(x)=2x45x310x2+21x4\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. x35x22x+10;2\begin{align*}x^3-5x^2-2x+10; -2\end{align*}
2. x4+7x3+6x232x32;4,1\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  5x53x4+2x3+x27x+3\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 f(x)=3x37x2+8x2\begin{align*}f(x)=3x^{3}-7x^{2}+8x-2\end{align*}

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

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