Exercise

A polynomial is given by $p\left(x\right)=x^3-2x^2+x$. For what values of $x$ is the polynomial $p\left(x\right)=0$?

If $\alpha$ and $\beta$ are the zeroes of polynomial $x^2+8x+6$. Form a polynomial whose zeroes are $\alpha -\beta \text{and} \alpha +\beta$.

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-3x-2$,  find a quadratic polynomial whose zeros are $\frac{1}{2\alpha +\beta }$ and $\frac{1}{2\beta +\alpha }$

If the zeroes of the cubic polynomial $x^3-6x^2+3x+10$ are of the form $a, a+b$ and $a+2b$ for some real numbers $a$ and $b$, find the values of $a$ and $b$ as well as the zeroes of the given polynomial.

If $a(70a-12b+2c)>0$ and the quadratic equation $a{x}^2+bx+c=0$ real roots $\alpha$ and $\beta$, then which of the following statements is always false?

Draw the graph of $y=x^2-4$ and hence solve $x^2-x-12=0$.

If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $k{x}^2+4x+4$ such that ${\alpha }^2+{\beta }^2=24$, then find the value of $k?$

Find the value of $k,$ such that one zero of the polynomial $3x^2+(1+4k)x+k^2+5$ is one third of other.