3.3: Polynomial Equations
- Page ID
- 238
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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.