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Polynomial Equations I. Homework II. Polynomial Equations 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: Step 1. 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. Step 2. Get rid of denominators by multiplying by the least common denominator. Step 3. If there is a common factor for all the terms, factor immediately. Otherwise, multiply the terms out. Step 4. Use a calculator to locate roots. Step 5. Use the Rational Root Theorem and synthetic division to exactly determine the roots. Example: Solve: (2x3 - 5)/4 = x - x2 1) (2x3 - 5)/4 - x + x2 = 0 2) (2x3 - 5)- 4x + 4x2 = 0 3) 2x3 + 4x2 - 4x - 5 = 0 4) From the graph, we see that there is a root between -3 and -2 and a root between 0 and -1 and a root between 1 and 2. 5) Since the only possible rational roots are 1,-1,5,-5,.5,-.5,2.5,-2.5, the possible rational roots are -5/2 and -.5. Neither of these two are roots, hence there are no rational roots. Example Solve x[x2(2x + 3) + 10x + 17] + 5 = 2 1) x[x2(2x + 3) + 10x + 17] + 3 = 0 3) 2x4 + 3x3 + 10x2 + 17x + 3 = 0 4) We see that there is a root between -2 and -1 and between -1 and 0. 5) Our only possible roots are -1/2 and -3/2 6) Using synthetic division, we see that -3/2 is a root, and the remainder is 2x3 + 10x + 2 = x3 + 5x + 1 which has no rational roots. Hence the rational root is -3/2 and using the calculator we see that the real root is .198.
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