Solve the equation x³− 9x²+14x + 24 = 0 if it is given that two of its roots are in the ratio 3:2.

If α, β, and γ are the roots of the polynomial equation ax³+ bx²+ cx + d = 0, find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must  be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

Find a polynomial equation of minimum degree with rational coefficients, having 2-\(\sqrt{3}\) as a root.

Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.

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

If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+ 62x⁴-126x³+ 65x²+127x-140 = 0, find all roots.

Find the condition that the roots of ax³+ bx²+ cx + d = 0 are in geometric progression. Assume a, b, c, d ≠ 0.

Solve the equation (x-2) (x-7) (x-3) (x+2)+19 = 0

Solve the equation (2x-3) (6x-1) (3x-2) (x-2)-5 = 0

Solve : (x - 5) (x - 7) (x + 6) (x + 4) = 504

Solve: (2x-1) (x+3) (x-2) (2x+3)+20 = 0

Solve the equation 9x³- 36x²+ 44x -16 = 0 if the roots form an arithmetic progression.

Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.

Find all zeros of the polynomial x⁶- 3x⁵- 5x⁴ + 22x³- 39x²- 39x + 135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.

Solve the equation : x⁴-14x² + 45 = 0

Solve the following equation: x⁴-10x³+ 26x²-10x + 1 = 0

Solve the equations:
6x⁴- 35x³+ 62x²- 35x + 6 = 0

Solve the equation 6x⁴- 5x³- 38x²- 5x + 6 = 0 if it is known that \(\frac{1}{3}\) is a solution.

Show that the equation x⁹- 5x⁵+ 4x⁴+ 2x²+ 1 = 0 has atleast 6 imaginary solutions.

Solve: (x - 4)(x - 7)(x - 2)(x + 1) = 16

Solve the equations
x⁴+ 3x³- 3x - 1 = 0