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 a value, a, is a root, then f(a)
0. (Roots are solutions to an equation 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 factor that
we know into the original polynomial
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Find the Other Roots (cont)
The resulting polynomial is a quadratic, but it
doesnt have real factors. Solve the quadratic
set equal to zero by either using the quadratic
formula, or by isolating the x and taking the
square root of both sides.
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Find the Other Roots (cont)
The solutions to the quadratic equation
For the polynomial
The three complex roots of the polynomial are
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More Practice