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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Roots & Zeros of Polynomials III Using the Rational Root Theorem to Predict the Rational Roots of a Polynomial Created by K. Chiodo, HCPS

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:

The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

List the Possible Rational Roots For the polynomial: All possible values of: All possible Rational Roots of the form p/q:

Narrow the List of Possible Roots For the polynomial: Descartes’ Rule: All possible Rational Roots of the form p/q:

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!)

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:

Find the Other Roots (con’t) The resulting polynomial is a quadratic, but it doesn’t 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.

Find the Other Roots (con’t) The solutions to the quadratic equation: The three complex roots of the polynomial are: For the polynomial:

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