Fundamental Theorem of Algebra TS: Demonstrating understanding of concepts

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Fundamental Theorem of AlgebraTS Demonstrating
understanding of concepts

• Warm-Up
• T or F A cubic function has at least one real
  root.
• T or F A polynomial function can have no
  complex solutions.
• T or F A polynomial function could have only
  one imaginary solution.
• T or F A polynomial could have root 2 as its
  only irrational solution.

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The Fundamental Theorem of Algebra

• If f(x) is a polynomial of degree n, where n gt 0,
  then f has at least one zero in the complex
  number system.
• Linear Factorization Theorem
• If f(x) is a polynomial of degree n where n gt 0,
  f has precisely n linear factors
• f(x) an(x c1)(x c2)(x cn)
• where c1, c2, , cn are complex numbers.

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Find a cubic polynomial with zeros of 2i and 3
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Find the quartic polynomial with zeros -v2 and i,
which passes through (1, 6)
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Factoring Polynomials so they are irreducable
over the rationals, reals and complex zeros.

• Factor each
• a) x4 x2 20

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Factoring Polynomials so they are irreducable
over the rationals, reals and complex zeros.

• Factor each
• x4 3x3 x2 12x 20
• (Hint x2 4 is a factor)

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You Try1) If -1 3i is a zero of x3 4x2
14x 20, find the other zeros

• 2) Factor the following x4 6x2 27
• a) Irreducible over the rationals
• b) Irreducible over the reals
• c) Irreducible over the complex