Question

Solve the equation $x^5-5x^4-5x^3+25x^2+4x-20=0$, whose roots are of the form $a,-a,b,-b,c$.

Answer 1

Given equation, $x^5-5x^4-5x^3+25x^2+4x-20=0$ with the roots $a,-a,b,-b,c$

$a-a+b-b+c=5⟹c=5\dots \dots \dots \dots \dots \dots \dots .\left(1\right)$

$-a^2+ab-ab+ac-ab+ab-ac-b^2+bc-bc=-5⟹a^2+b^2=5\dots \dots \dots \dots \dots \dots ..\left(2\right)$

$a^2{b}^{2}c=20⟹a^2{b}^2=4\dots \dots \dots \dots \dots \dots .\left(3\right)$

Eliminating $b$ in (2) using (3), we have

$a^2+\frac{4}{a^2}=5⟹a^4–5a^2+4=0$

Solving the quadratic, we get $a^2=4\text{or}{a}^2=1$

$\therefore \text{the roots are}-1,1,-2,2,5$