Relationship between Zeroes and Coefficients of a Polynomial

If $\sqrt{2}$ is a zero of the cubic polynomial $6x^3+\sqrt{2}{x}^2-10x-4\sqrt{2}$, then find its other two zeroes.

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.

Find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of this polynomial by factorisation.

$\frac{-3}{2\sqrt{5}},-\frac{1}{2}$

Find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of this polynomial by factorisation.

$-2\sqrt{3},-9$

Find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of this polynomial by factorisation.

$\frac{21}{8},\frac{5}{16}$

Find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of this polynomial by factorisation.

$\frac{-8}{3},\frac{4}{3}$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$7y^2-\frac{11}{3}y-\frac{2}{3}$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$y^2+\frac{3}{2}\sqrt{5}y-5$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$v^2+4\sqrt{3}v-15$.

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$2s^2-\left(1+2\sqrt{2}\right)s+\sqrt{2}$.

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$4x^2+5\sqrt{2}x-3$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$2x^2+\frac{7}{2}x+\frac{3}{4}$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$t^3-2t^2-15t$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$5t^2+12t+7$

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials

$3x^2+4x-4$

Find the zeroes of the following polynomial by factorisation method and verify the relations between the zeroes and the coefficients of the polynomial

$4x^2-3x-1$

The only value of $k$ for which the quadratic polynomial $k{x}^2+x+k$ has equal zeroes is $\frac{1}{2}$.

If all three zeroes of a cubic polynomial $x^3+a{x}^2-bx+c$ are positive, then atleast one of $a, b$ and $c$ is non-negative.

If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.

If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.