7'5 Roots and Zeros

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7.5 Roots and Zeros

• As Taught by Grace Ann Whiteside

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Objective One

• Determining the number and type of roots
• for a polynomial equation.
• The leading coefficient tells you how many roots
  a polynomial equation will have.
• X 3 0 will just have one. X2 8x 160 will
  have two. You know this because of the x2.

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Objective One (cont)

• There are four possible outcomes for roots.
• Real, imaginary, positive real, and negative real.

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Solving X3 2x 0

• The first method you want to try in solving the
  equation is factoring.
• X3 2x 0
• x(x2 2) 0
  factor out GFC
• Use Zero Product Property
• x 0 or x2 2 0
• x2 -2
• x /_

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Remember complex numbers are both real and
imaginary!
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Fundamental Theorem of Algebra

• A polynomial equation with degree gtzero, it may
  have one or more real roots, or no real roots
  (roots are imaginary numbers). Keep in mind that
  real and imaginary numbers belong to the same set
  of complex number (5 2i)
• All polynomial expressions with degree gtzero will
  have at least one root in the set of complex
  numbers.

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Polynomial Equations

• A polynomial equation is set to equal zero.
• P(x) 0
• Example P(x) 5x3 4x2 7x 8 0

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Roots and Zeros

• Roots and Zeros are a solution to the polynomial
  equation P(x) 0
• That is, the number r is the root of a polynomial
  P(x) 0 if and only if P(r) 0.

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Descartes Rule of Signs

• Imaginary solutions always occur in conjugate
  pairs.
• If a bi is a root, then a bi is also a root.
• The number of positive REAL zeros of y P(x) is
  the same as the number of changes in sign of the
  coefficients of the terms, or is less by an even
  number. Same for negative real zeros(P(-x))

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Example of Descartes Rule

• X3 4x2 6x 4
• Hint look at the changes from sign to sign.
• A positive x3 to a negative 4x2 is counted as a
  change.
• There are 3 or 1 positive real zero, 0 negative
  real zero, or all 3 are positive real zeros.
• - 1 positive 2 are imaginary
• 2 1 /- i

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Example

• (-x)5 6(-x)4 3 (-x)3 7(-x)2 8 (-x) 1
• -x5 6x4 3x3 7x2 8x 1
• There was only one change from sign to sign.
• There are 5 roots. 1 neg 4 pos.
• Options 4 pos real roots, 2 pos real roots w/2
  imaginary, 0 pos w/4 imaginary.
• 1 neg w/1 neg, 2 pos w/0 pos, 2 imaginary w/4
  imaginary.

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• END