3.3: Polynomial Equations

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So far we have learned how to find the roots of a polynomial equation. If we have an equation that involves only polynomials we follow the steps:

Bring all the terms over to the left hand side of the equation so that the right hand side of the equation is a 0.
Get rid of denominators by multiplying by the least common denominator.
If there is a common factor for all the terms, factor immediately. Otherwise, multiply the terms out.
Use a calculator to locate roots.
Use the Rational Root Theorem and synthetic division to exactly determine the roots.

Since the only possible rational roots are 1, -1, 5, -5, .5, -.5, 2.5, -2.5, the possible rational roots are \(-\dfrac{5}{2}\) and -.5. Neither of these two are roots, hence there are no rational roots.

\[2x^3 + 10x + 2 = 2(x^3 + 5x + 1)\]

which has no rational roots. Hence the rational root is \(-\frac{3}{2}\) and using the calculator we see that the irrational root is 0.198.

Larry Green (Lake Tahoe Community College)

Integrated by Justin Marshall.

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