Relationship between Zeroes and Coefficients of a Polynomial

If the sum of the squares of zeroes of polynomial $p\left(x\right)=x^2-8x+k$ is $40$. Find the value of $k$

Find the zeroes of $\sqrt{3}{x}^2+8x+5\sqrt{3}$ and verify the relation between the zeroes:

Find the zeroes of the polynomial $x^2+2\sqrt{2}x-6$  and verify the relationship between the zeroes and the coefficients.

Find the zeroes of the polynomial $6m^2-3$  and verify the relationship between the zeroes and the coefficients.

Find the zeroes of the polynomial $x^2+7x+12$  and verify the relationship between the zeroes and the coefficients.

Find the zeroes of  the polynomial $4\sqrt{3}{x}^2+5x-2\sqrt{3}$  and verify the relationship between the zeroes and the coefficients.

Find the zeroes of  the polynomial $6x^2-x-2$  and verify the relationship between the zeroes and the coefficients.

Find the quadratic polynomial whose zeroes are $3\sqrt{2}$ and $-2\sqrt{2}$ also write the coefficients of the polynomial.

Find a quadratic polynomial with zeroes $2$ and $\frac{-1}{3}$

If one zero of the polynomial $6x^2+37x-(k-2)$ is reciprocal of the other, If then what is the value of $k?$

If $p \text{and} q$ are the zeroes of the polynomial $3x^2-5x+2$, write the polynomial in $x$ whose zeroes are $\frac{1}{p}$ and $\frac{1}{q}$.

If one root of the quadratic equation $3x^2+px+4=0$ is $\frac{2}{3}$, then find the value of $p$ and another root of the equation.

The roots $\alpha \text{and} \beta$ of the quadratic equation $x^2-5x+3(k-1)=0$ are such that $\alpha -\beta =1$. Find the value of $k$

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p\left(x\right)=6x^2+x-12$, then find a quadratic polynomial whose zeroes are $\alpha +2$ and $\beta +2$.

If $a$ and $b$ are the zeroes of the polynomial $2x^2-4x+5$, then find the value of $a^2+b^2$

If $m$ and$n$ are zeroes of the polynomial $3x^2-x-2$, find the values of the following without factorising the polynomial.

(i) $\frac{1}{m}+\frac{1}{n}$

(ii) $m^2+n^2$

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Find the quadratic polynomial whose zeros are $-4$ and $\frac{3}{2}$.

The zeros of a quadratic polynomial are in the ratio of $2:3$ and their sum is $27$. Find the product of these zeros.

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $z^2-3z-18,$ then the value of $\left({\beta }^2{\alpha }^3+{\alpha }^2{\beta }^3\right)$ is $?$

Find a cubic polynomial, where zeroes are $\frac{2+\sqrt{5}}{2}, \frac{{\displaystyle 2-\sqrt{5}}}{{\displaystyle 2}}$ and $4$ respectively.