Exercise 3 Page 197

Let's start by reviewing how to solve an equation by graphing. We can solve a general polynomial equation, $a_n{x}^n$ $+a_{n-1}{x}^{n-1}$ $+\dots$ $+a_{1}x$ $+a_0,$ by graphing the related function $f(x)=$ $a_n{x}^n$ $+a_{n-1}{x}^{n-1}$ $+\dots$ $+a_{1}x$ $+a_0.$ Notice that we recover the original equation when $f(x)=0.$ Since the points on the graph have the form $(x,f(x))$ the real solutions will be the $\mathbf{x}\text{-intercepts}$ $(x,0).$ Since even degree polynomials have the same end behavior in both directions, there will be cases were the graph will not intersect the $x\text{-axis}$ at all. In these cases, all the solutions are imaginary.

Nevertheless, there are cases where a graph does not give enough information, or maybe it is just complicated to determine if there are imaginary solutions. Consider the function shown below.

In the example above, we have a polynomial function of degree $8.$ The related equation has $8$ real solutions even though the $x\text{-intercepts}$ are just $3.$ This is because the solution $x=0$ repeats $6$ times. This can be seen when we write the polynomial equation in factored form. For cases like this it is hard to determine if all the solutions are real, or if there are missing solutions which must then be imaginary. For these cases we cannot rely completely on the graphs, and we might need to use additional algebraic methods to determine whether there are imaginary solutions or not.