Relation Between Zeroes (Roots) and Coefficients of a Quadratic Equation

If α, β, γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, the\[\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} =\]

Can the quadratic polynomial x² + kx + k have equal zeroes for some odd integer k > 1?

Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.

`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

`(-8)/3, 4/3`

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha-1/beta`

If the zeroes of the polynomial x² + px + q are double in value to the zeroes of the polynomial 2x² – 5x – 3, then find the values of p and q.

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha+1/beta-2alphabeta`

If all three zeroes of a cubic polynomial x³ + ax² – bx + c are positive, then at least one of a, b and c is non-negative.