Vieta’s Formulae and Formation of Polynomial Equations

Solve the cubic equation $2x^3-x^2-18x+9=0$ if the sum of two roots vanishes.

A $12$ metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was cut away.

Formulate into a mathematical problem to find a number such that when its cube root is added to it, the result is $6$.

Solve the equation $3x^3-16x^2+23x-6=0$ if the product of two roots is equal to $1$.

If the equations $x^2+px+q=0$ and $x^2+p\text{'}x+q\text{'}=0$ have a common root, show that it must be equal to $\frac{p{q}^{\mathrm{\prime }}-p^{\mathrm{\prime }}q}{q-q^{\mathrm{\prime }}}$ or $\frac{q-q^{\mathrm{\prime }}}{p^{\mathrm{\prime }}-p}$.

If $p$ and $q$ are the roots of the equation $l{x}^2+nx+n=0$, then show that $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$

If $\alpha ,\beta ,\gamma$ and $\delta$ are the roots of the polynomial equation $2x^4+5x^3-7x^2+8=0$, find a quadratic equation with integer coefficients whose roots are $\alpha +\beta +\gamma +\delta$ and $\alpha \beta \gamma \delta$.

If $\alpha ,\beta$ and $\gamma$ are the roots of the polynomial equation $a{x}^3+b{x}^2+cx+d=0$, find the value of $\sum _{}^{}\frac{\alpha }{\beta \gamma }$ in terms of the polynomial coefficient.

Solve the equation $x^3-9x^2+14x+24=0$ if it is given that two of its roots are in the ratio $3:2$.

Find the sum of squares of roots of the equation $2x^4-8x^3+6x^2-3=0$.

If $\alpha ,\beta$ and $\gamma$ are the roots of the cubic equation $x^3+2x^2+3x+4=0$, form a cubic equation whose roots are $-\alpha ,-\beta$ and $-\gamma$.

If $\alpha ,\beta$ and $\gamma$ are the roots of the cubic equation $x^3+2x^2+3x+4=0$, form a cubic equation whose roots are $\frac{1}{\alpha },\frac{1}{\beta }$ and $\frac{1}{\gamma }$.

If $\alpha ,\beta$ and $\gamma$ are the zeros of $x^3+p{x}^2+qx+r=0$, then $\sum _{}^{}\frac{1}{\alpha }$ is 

If $\alpha ,\beta$ and $\gamma$ are the roots of the cubic equation $x^3+2x^2+3x+4=0$, form a cubic equation whose roots are $2\alpha ,2\beta$ and $2\gamma$.

Construct a cubic equation with roots $2,\frac{1}{2}$ and $1$.

Construct a cubic equation with roots $1,1$ and $-2$.

Construct a cubic equation with roots $1,2$ and $3$

If the sides of a cubic box are increased by $1$, $2$, $3$ units respectively to form a cuboid, then the volume is increased by $52$ cubic units. Find the volume of the cuboid.

(write only numerical value).