Finding solutions of polynomial equations where the degree of the resulting polynomial is higher than 2 is best done using graphing or CAS technology. However, if there is reason to believe that the solutions are integers, (or the resulting equation is a quadratic) the factor theorem may be used.
Find all values of such that
Graphing two functions together, let and as shown below.
We see that when , both curves have the same value.
This is also true when :
Therefore the expressions and have equal value when or when .
To solve algebraically, we reduce once side to zero by subtracting the terms of one side from both sides:
Another way to understand this algebra graphically is to graph the related function , that is,
Notice that the -intercepts of the graph have the same values as the intersections of the graphs and . That is because when we subtract two equal values, the answer is zero.
Because ; . In the same way, .
To solve an equation , we may
- Draw the graphs of both and locate the intersections of the graphs. We are looking for the values of the intersections.
- Draw the graph and locate the roots of the graphs.
For these two methods technology speeds up the process considerably. Here’s a how to video for this calculation on GeoGebra. Most graphing software has a tool or command for finding intersections and for finding -intercepts.
- To solve the equation algebraically, we first rearrange to , then solve by factoring if possible.
The equations in this applet all have at least one integer solution. If the resulting polynomial has a degree greater than 2, the factor theorem should be used to find the first solution.