Ch 2'5: The Fundamental Theorem of Algebra

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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

• 1. Find possible number of positive and negative
  zeros
• 2. Graph
• 3. Find the zeros
• x1 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
3

• 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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• 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