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Real Zeros of Polynomials

Remainder, Factor, and Rational Zero Theorems and Descartes' Rule of Signs.

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Practice Real Zeros of Polynomials
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Estimated19 minsto complete
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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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