H S Hall and S R Knight Solutions for Chapter: Theory of Equations, Exercise 4: EXAMPLES. XXXV. d.

If $a, b, c$ are the roots of the equation $x^3+qx+r=0$ form the equation whose roots are $\frac{b+c}{a^2}, \frac{c+a}{b^2}, \frac{a+b}{c^2}$.

If $a, b, c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are $bc+\frac{1}{a}, ca+\frac{1}{b}, ab+\frac{1}{c}$.

If $a, b, c$ are the roots of $x^3+qx+r=0$, form an equation, whose roots are $a(b+c), b(c+a), c(a+b)$.

If $a, b, c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are $a^3, b^3, c^3$.

If $a, b, c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are $\left(\frac{b}{c}+\frac{c}{b}\right), \left(\frac{c}{a}+\frac{a}{c}\right), \left(\frac{a}{b}+\frac{b}{a}\right)$.

Show that the cubes of the roots of $x^3+a{x}^2+bx+ab=0$ are given by the equation $x^3+a^3{x}^2+b^{3}x+a^3{b}^3=0$.

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

If the roots of $x^3+3p{x}^2+3qx+r=0$ are in harmonic progression; show that $2q^3=r\left(3pq-r^2\right)$.