Chapter 6.6-6.9 Test Outline

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6.6: factoring polynomials

  • n terms
    • greatest common factor
  • 2 terms
    • difference of squares
    • difference of cubes
    • sum of cubes
  • 3 terms
    • perfect square trinomials
    • general trinomials
  • 4 or more terms: grouping
  • using these in combination

6.7: remainder and factor theorems

  • synthetic substition: f(a) = f(x) / (x - a)
  • remainder theorem: f(x) / (x-a) has remainder f(a)
  • factor theorem: (x - a) is a factor of f(x) iff f(a) = 0

6.8: roots and zeros

  • Let f(x) be a polynomial function. Then:
    • c is a zero of f(x)
    • x-c is a factor of f(x)
    • c is a root or solution of f(x) = 0
    • (c, 0) is an x-intercept of f(x)
  • fundamental theorem of algebra
    • every polynomial of degree 1 or greater has at least one root in the complex numbers
    • corollary: a polynomial of degree n has n roots in the complex numbers
  • Descartes rule of signs
  • complex conjugates theorem
    • complex roots come in conjugate pairs
  • use these to find all roots
  • write polynomials using roots

6.9: rational zero theorem

  • find all possible rational roots
  • find rational zeros
  • find remaining zeros
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