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**Ch 2.5: The Fundamental Theorem of Algebra**

Theorem: In the complex number system, every nth degree polynomial has n zeros Complex system: Real AND Imaginary numbers Finding all zeros 1. Use Descartes to determine possible solutions 2. Use the calculator to find all rational zeros 3. Use synthetic division to get the problem to a quadratic equation 4. Use the quadratic formula to find the remaining zeros ***If a solution is imaginary, then its conjugate is a solution as well***

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**Ex: Total Solutions: 5 Positive: 2 or 0 Negative: 1 Imaginary: 4 or 2**

1. Find possible number of positive and negative zeros 2. Graph 3. Find the zeros **x=1 has multiplicity of 2 to give 2 positive zeros** Continued…. Positive: 2 or 0 Negative: 1 Imaginary: 4 or 2 X = -2 X = 1 X = 1

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**ALL SOLUTIONS 5. Synthetic divide one zero at a time**

6. Use the quadratic formula to find the final zeros ALL SOLUTIONS

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**Sometimes you must factor…**

1. Graph give 2 irrational zeros, so try factoring! 2. Set each factor equal to zero ALL SOLUTIONS

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**Find all roots when given one complex root**

1. Change it to a factor 2. Multiply it by its conjugate (because it is a root as well) 3. Use Long division to get the remaining quadratic equation 4. Factor the equation to find the zeros or use the quadratic formula

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**Ex: Find all the roots of if 1 + 3i is a zero**

1. Turn it into a factor 2. Multiply by its conjugate 3. Divide by the new factor Continued…

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**(remember, imaginary and its conjugate)**

4. Factor the quotient ALL SOLUTIONS (remember, imaginary and its conjugate)

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**Ex: Find the 4th degree polynomial with zeros of 1, 1, and 3i**

Turn into factors Remember conjugate is a factor as well!! Multiply and Simplify!

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**Ex: Find the cubic with zeros, 2 and 1-i where f(1)=3**

1. Write the factors, including the conjugate 2. To find f(1) = 3, take the function, put a(function)=3 and solve 3. Plug in 1 for x and find a 4. Distribute a

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