Complex Zeros and the Fundamental Theorem of Algebra

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Complex Zeros and the Fundamental Theorem of
Algebra

• 2.6

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Ccomplex numbersabi or bi or a Iimaginary
numbers--bi Rreal numbersa
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Whole0,1,2,3,4, Integers---3,-2,-1,0,1,2,3 Rat
ional Numbersp/q Irrational Numbersp, e, square
root of primes
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The Fundamental Theorem of Algebra

• A polynomial function of degree ngt0 has n complex
  zeros. Some of the zeros may be repeated.

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Linear Factorization Theorem

• If f(x) is a polynomial function of degree ngt0,
  then f(x) has precisely n linear factors and
• F(x) a(x-z1)(x-z2)(x-zn)
• Where a is the leading coefficient of f(x) and
  z1, z2,, zn are the complex zeros of f(x). Each
  z is not necessarily distinct, as a zero can be
  repeated.

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Equivalent Statements in the Complex Number World.

• xk is a solution or root of the equation f(x)
  0.
• k is a zero of the function f.
• x-k is a factor of f(x).

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Equivalent Statements in the Real Number World.

• xk is a solution or root of the equation f(x)
  0.
• k is a zero of the function f.
• x-k is a factor of f(x).
• k is an x-intercept of the graph of yf(x)

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Complex Conjugate Zeros

• Suppose that f(x) is a polynomial function with
  real coefficients. If a and b are real numbers
  with b not equal zero and abi is a zero of f(x),
  then its complex conjugate a-bi is also a zero of
  f(x).

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Factors of a Polynomial with Real Coefficients

• Every polynomial function with real coefficients
  can be written as a product of linear factors and
  irreducible quadratic factors, each with real
  coefficients.

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Polynomial Function of Odd Degree

• Every polynomial function of odd degree with real
  coefficients has at least one real zero.

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Write a polynomial in standard form.

• If a complex zero exists, then its complex
  conjugate is also a zero.
• Change the zeros to factors.
• If necessary, place a power on each factor based
  on multiplicity.
• Foil.

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Choosing a Graph of a Polynomial without a
Calculator

• Determine the multiplicity of the zero.
• Use that to determine if there is a cross or a
  touch.
• Choose the graph that best fits.

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State the number of Complex and Real Zeros

• The number of complex zeros will be equal to the
  degree of the polynomial.
• Graph the polynomial to determine the number of
  real zeros. You may have to zoom out to find all
  of the real zeros.
• Remember that complex zeros come in conjugate
  pairs, so odd degrees need at least one real zero.

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Find all zeros and write all the linear factors
of the polynomial

• Use the calculator and the rational zeros theorem
  to find all of the real zeros.
• Use synthetic division on one of the real zeros.
• Use synthetic division again with another real
  zero using the results from the first.
• Repeat until a quadratic equation is found.

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Find all zeros and write all the linear factors
of the polynomial

• Use quad form on the results of the second
  synthetic division.
• Write each zero as a factor.

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Find all zeros and write all the linear and
irreduciable factors of the polynomial

• Follow the previous instructions, but when you
  get a quadratic, only use quad form if the
  results will be real, otherwise leave it as a
  quadratic.
• Use the discriminant to help determine whether or
  not to factor the quadratic.