A polynomial equation is an equation that contains a polynomial expression. An example is By rearranging the equation so that one side is it's possible to identify the degree of the equation. The equation above can be rewritten asThe maximum number of solutions of an equation is given by the degree. Since the highest exponent is this equation has a maximum of 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.
Solve the equation by factoring.
Similar to other equations, polynomial equations can be solved graphically. Consider the following equation. Usually, all terms are gathered to one side to create an equivalent equation. The polynomial expression on the left-hand side can be viewed as the function The solutions to the original equation are the zeros of the function. Graphing the function allows the zeros to be seen easily.
Sketch the following polynomial function using zeros and end behavior.
The coefficient in the -term is positive, meaning that when approaches infinity, the graph extends upward.
Since the function is a third-degree polynomial — odd — the left end extends in the opposite direction: downward.
Lastly, let's sketch the rest of the graph. We don't know exactly how it looks, but it intersects the -axis at and
Note that other sketches are possible, as long as the zeros and end behavior are the same.