Polynomial Equations

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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:

Find all the rational solutions of

(2x³ - 5)/4 = x - x²

(2x³ - 5)/4 - x + x² = 0
(2x³ - 5)- 4x + 4x² = 0
2x³ + 4x² - 4x - 5 = 0
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.
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

$Graph that crosses the x-axis between x=-2 and x=-1, then between x=-1 and x=0.$

x[x²(2x + 3) + 10x + 17] + 3 = 0
2x⁴ + 3x³ + 10x² + 17x + 3 = 0
We see that there is a root between -2 and -1 and between -1 and 0.
Our only possible roots are -1/2 and -3/2.
Using synthetic division, we see that -3/2 is a root, and the remainder is

2x³ + 10x + 2 = 2(x³ + 5x + 1)

which has no rational roots. Hence the rational root is -3/2 and using the calculator we see that the irrational root is 0.198.

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