Polynomial Equations

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.

Example

Find all values of x such that

    \[2x^2-x-2=4x+5\]

Graphing two functions together, let f(x)=2x^2-x-2 and g(x)=4x+5 as shown below.

We see that when x=-1, both curves have the same y value.

    \[f(-1)=2(-1)^2-(-1)-2=1;\quad \quad g(-1)=4(-1)+5=1\]

This is also true when x=3.5:

    \[f(3.5)=2(3.5)^2-3.5-2=19; \quad \quad g(3.5)=4(3.5)+5=19\]

Therefore the expressions 2x^2-x-2 and 4x+5 have equal value when x=-1 or when x=3.5.

To solve algebraically, we reduce once side to zero by subtracting the terms of one side from both sides:

    \begin{align*}2x^2-x-2&=4x+5\\[10 pt] 2x^2-5x-7&=0\\[10 pt] (2x-7)(x+1)&=0\\[10 pt]2x-7 = 0 \quad \text{or}&\quad x+1=0\\[10 pt]x=\dfrac{7}{2} \quad \text{or}&\quad x = -1\end{align}

Another way to understand this algebra graphically is to graph the related function h(x)=f(x)-g(x), that is,

    \[h(x)=(2x^2-x-2)-(4x+5)=2x^2-5x-7\]

Notice that the x-intercepts of the graph h(x) have the same x values as the intersections of the graphs f(x) and g(x). That is because when we subtract two equal values, the answer is zero.

Because f(-1)=g(-1); f(-1)-g(-1)=1-1=0. In the same way, f(3.5)-g(3.5)=19-19=0.

Conclusion

To solve an equation f(x)=g(x), we may

  • Draw the graphs of both and locate the intersections of the graphs. We are looking for the x values of the intersections.
  • Draw the graph h(x)=f(x)-g(x) 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 x-intercepts.

  • To solve the equation f(x)=g(x) algebraically, we first rearrange to f(x)-g(x)=0, then solve by factoring if possible.

Practice

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.