Basics of Polynomial Equations and its Formation

A quadratic polynomial$y = f\left(x\right)$ satisfies $f\left(x\right)={\left[\frac{f\left(x+1\right)-f\left(x-1\right)}{2}\right]}^2$ for all real x. then the value of $\left[f\left(0\right)-f\left(-1\right)\right]+\left[f\left(0\right)-f\left(1\right)\right]$ is

If one root of the equation  $x^2+px+q=0$ is square of the other root, then

Prove that a line cannot intersect a parabola at more than two points.

Prove that a line cannot intersect a rectangular hyperbola at more than two points.

Prove that a line cannot intersect a hyperbola at more than two points.

Prove that a line cannot intersect an ellipse at more than two points.

Prove that a line cannot intersect a circle at more than two points.

The denominator of our possible solutions from the rational roots theorem comes from the factors of our constant term
in our polynomial.

The denominator of our possible solutions from the rational roots theorem comes from which term in our polynomial?

The numerator of our possible solutions from the rational roots theorem comes from which term in our polynomial?

Describe the Rational Root Theorem.

Show that the polynomial $x^9+9x^7+7x^5+5x^3+3x+2$ has at least eight imaginary roots.

Show that $x^9-5x^8-14x^7+2$ has at least six imaginary roots.

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

Show that $x^5-19x^4+2x^3+5x^2+11$ has at least two imaginary roots.

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

Solve the equation $8x^3-36x^2+22x+21=0$ if the roots form an arithmetic progression.

Solve the equation $4x^3-24x^2+23x+18=0$ if the roots form an arithmetic progression.

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 $x^5+11x^4+x^3+4x^2-13x+6=0$.