Question

The equation $x^5-209x+56=0$ has two roots whose product is unity : determine them.

Answer 1

Denote the roots by ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{\frac{1}{a}}^{\prime }$ then we have $a^5-209a+56=0$

and $56a^5-209a^4+1=0$

Eliminating $a^5$, we have

$\left(209\right)a^4-209a\times 56+{56}^2-1=0$

$\Rightarrow a^4-56a+15=0$

Substituting by eliminating the constant from the two above equations and dividing by a

We have $15a^4-56a^3+1=0$

$\Rightarrow$ From these last two equations, we find

$a^3-15a+4=0$ and $4a^3-15a^2+1=0$

Now eliminating $a^3$ we have

$a^2-4a+1=0$

$a=2\pm \sqrt{3}$