Show that the equation cannot have imaginary roots

Question

Show that the equation $$ \dfrac{a}{x-1} + \dfrac{b}{x-2} + \dfrac{c}{x-3} + \dfrac{d}{x-4} = 1 $$ cannot have imaginary roots if $a, b, c, d$ are any four real numbers of the same sign. I tried putting $x = a$ and some other variables to show that the equation changes sign 3 times but failed.

Answer 1

The resulting fourth degree equation has necessarily 4 real roots.

Here is a convincing reason, followed by a proof.

Have a look at the graphics below. It is conversion of the purely algebraic issue into a search for the number of intersection points between the curves of functions $f$ and $g$ with the following equations:

$$\begin{cases}y=f(x)=\frac{0.1}{x-1}+\frac{0.3}{x-2}+\frac{0.5}{x-3}+\frac{0.2}{x-4}\\y=g(x)=1\end{cases}$$

The rigorous proof comes from the fact that, assuming all coefficients to be positive, $f'(x)<0$, thus $f$ is decreasing on all intervals where it is defined, i.e.,

$$(-\infty,1),(1,2),(2,3),(3,4),(4,\infty).$$

A thorough inspection of the last four intervals show that there is an intersection point in all of them.

For example, the existence of a unique intersection point in interval $(1,2)$,is due to the fact that $f$ is decreasing and that

$$\cases{\lim_{x \rightarrow 1_+}f(x)=+\infty\\ \lim_{x \rightarrow 2_-}f(x)=-\infty}$$

For interval $(4,+\infty)$, it is because $f$ is decreasing and:

$$\cases{\lim_{x \rightarrow 4+}f(x)=+\infty\\ \lim_{x \rightarrow \infty}f(x)=0_+}$$