Question 48

Denote the wives by $A$, $B$, $C$, and $D$, and their respective husbands $a$, $b$, $c$, and $d$.

First we seat the wives.

The first wife has a choice of $8$ seats, but this does not lead to different seating arrangements as we are only interested in where the guests are sitting relative to each other.

Once the first wife is seated there are $3\times2\times1=6$ possibilities for the other $3$ wives.

Thus the $4$ wives may be seated in $6$ different ways.

For each arrangement of wives there are only $2$ possibilities for the husbands, as shown.

Hence there are $6\times2=12$ different ways of seating the couples.

Thus, $\sqrt{5}-\sqrt{2}$ is a root of $x^{4}-14x^{2}+9=0$.

By the “theorem on rational roots”, the only possible rational roots are $\pm 1$, $\pm 3$, and $\pm 9$.

$\sqrt{5}-\sqrt{2}$ is not equal to any of these values.
$\enspace\therefore\quad\sqrt{5}-\sqrt{2}$ is not a rational root.
$\enspace\therefore\quad\sqrt{5}-\sqrt{2}$ is irrational.