Roots

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Roots Zeros of Polynomials III

• Using the Rational Root Theorem to Predict the
  Rational Roots of a Polynomial

Created by K. Chiodo, HCPS
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Find the Roots of a Polynomial
For higher degree polynomials, finding the
complex roots (real and imaginary) is easier if
we know one of the roots.
Descartes Rule of Signs can help get you
started. Complete the table below
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The Rational Root Theorem
The Rational Root Theorem gives us a tool to
predict the values of rational roots
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List the Possible Rational Roots
For the polynomial
All possible values of
All possible rational roots of the form p/q
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Narrow the List of Possible Roots
For the polynomial
Descartes Rule
All possible Rational Roots of the form p/q
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Find a Root That Works
For the polynomial
Substitute each of our possible rational roots
into f(x). If f(a) 0, then a is a root of
f(x). (Roots are the solutions to the polynomial
set equal to zero!)
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Find the Other Roots
Now that we know one root is x 3, do the other
two roots have to be imaginary? What other
category have we left out?
To find the other roots, divide the root we know
into the original polynomial
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Find the Other Roots (cont)
The degree of the resulting polynomial is 1 less
than the original polynomial. When the resulting
polynomial is a QUADRATIC, we can solve it by
FACTORING or by using the QUADRATIC FORMULA!
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Find the Other Roots (cont)
This quadratic does not have real factors, but it
can be solved easily by moving the 5 to the other
side of the equation.
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Find the Other Roots (cont)
The roots of the polynomial equation
are
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Another Example

• Descartes rule of signs Total Roots 4
• 
  Real Roots 1
• -
  Real Roots 3 or 1
• 
  Imag. Roots 0 or 2
• The possible RATIONAL roots

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Another Example

• Find a Rational Root that works
• f(2) 0, so x 2 is a root
• Synthetic Division with the root, x 2

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Another Example

• The reduced polynomial is

• Since the degree gt 2, we must do synthetic
  division again, go back to the list of possible
  roots and try them in the REDUCED polynomial.
  The same root might work again - so try it also!
• f(-1/3) 0, so x - 1/3 is a
  root

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Another Example

• Synthetic Division with the root, x - 1/3 into
  the REDUCED POLYNOMIAL

• The reduced polynomial is now a quadratic

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Another Example

• Solve the resulting quadratic using the quad.
  formula

• The 4 roots of the polynomial are

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More Practice
For each of the polynomials on the next page,
find the roots of the polynomials.

• Know what to expect ---- Make a table using
  Descartes Rule of Signs.
• List the possible RATIONAL roots, p/q.
• Find one number from your p/q list that makes the
  polynomial 0. You can check this either by
  evaluating f() or by synthetic division.
• Do synthetic division with a root - solve the
  resulting polynomial.

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More Practice
Find the roots of the polynomials