# Solving Polynomial Equations

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Polynomials are usually written in standard form. However, depending on what they describe and what information is needed, it's sometimes useful to write them in factored form. By rewriting polynomials as products of their factors, it's possible to solve polynomial equations.

## Polynomial Equation

A polynomial equation is an equation that contains a polynomial expression. An example is $x^3 - x = 7x^2 + 3.$ By rearranging the equation so that one side is $0,$ it's possible to identify the degree of the equation. The equation above can be rewritten as $x^3 - 7x^2 - x - 3 = 0.$

The maximum number of solutions of an equation is given by the degree. Since the highest exponent is $3,$ this equation has a maximum of $3$ solutions. In order to solve some polynomial equations, algebraic methods, such as the Quadratic Formula and the Zero Product Property, can be used. Alternatively, a graphic solution works for any polynomial equation and numerical methods may also be used.## Solving a Polynomial Equation Graphically

Similar to other equations, polynomial equations can be solved graphically. Consider the following equation. $x^3-2x^2-2x+1=2x^2-3x-5$ Usually, all terms are gathered to one side to create an equivalent equation. $x^3-4x^2+x+6=0$ The polynomial expression on the left-hand side can be viewed as the function $y=x^3-4x^2+x+6.$ The solutions to the original equation are the zeros of the function. Graphing the function allows the zeros to be seen easily.

They are $x=\text{-}1,$ $x=2,$ and $x=3.$ Thus, the solutions to the original equation are $x=\text{-}1,$ $x=2,$ and $x=3.$ By substituting the solutions into the equation, it's possible to determine if the solutions are exact or approximate.## Exercises

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