Polynomial Equation and Identity

Show that $9x^9+2x^5-x^4-7x^2+2$ has at least six imaginary roots.

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-px+q$, then find the values of ${\alpha }^2+{\beta }^2$ and $\frac{1}{\alpha }+\frac{1}{\beta }$.

Find the polynomial equation whose roots are the translates of the roots of the equation $3x^5-5x^3+7=0$ by $4$.

Find the polynomial equation whose roots are the translates of the roots of the equation $x^4-x^3+10x^2+4x+24=0$ by $2$.

Find the polynomial equation whose roots are the translates of those of the equation $x^5-4x^4+3x^2-4x+6=0$ by $-3$.

Find the polynomial equation whose roots are the translates of those of the equation $x^4-5x^3+7x^2-17x+11=0$ by $-2$.

Find the algebraic equation of degree $4$ whose roots are the translates of the roots of $4x^4+32x^3+83x^2+76x+21=0$ by $2$.

Show that $2x^3+5x^2+5x+2=0$ is reciprocal equation of class one.

Find the polynomial equation whose roots are the reciprocals of the roots of $x^5+11x^4+x^3+4x^2-13x+6=0$.

Find the polynomial equation whose roots are the reciprocals of the roots of $x^4-3x^3+7x^2+5x-2=0$.

Find the polynomial equation whose roots are the reciprocals of the roots of the equation $x^4+3x^3-6x^2+2x-4=0$.

Solve the equation $x^2-\left(3\sqrt{2}-2i\right)x-6\sqrt{2}i=0$ by using general expressions for the roots of the equation.

Solve the equation $3x^2+7ix+6=0$ by factorization method.

Solve the equation $x^2-\sqrt{2}ix+12=0$ by factorization method.

Solve the equation $x^2-4x+13=0$ by factorisation method.

The equation $\frac{a^2}{x-a^{\text{'}}}+\frac{b^2}{x-b}+\frac{c^2}{x-c^2}+\dots +\frac{k^2}{x-k^{\text{'}}}=x-m,$ where $a,b,c,\dots ,k,m,a^{\text{'}},b_j,\dots ,k^{\text{'}}$ are all real numbers then 

Form the polynomial equation with rational coefficients whose roots are $4\sqrt{3},5+2i$.

Form the polynomial equation, whose roots are $1+i,1-i,1+i,1-i$. 

Form the polynomial equation, whose roots are $1+i, 1-i,-1+i,-1-i$. 

Form the polynomial equation, whose roots are $3,2,1+i, 1-i$.