Construct a cubic equation with roots 1, 2 and 3

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

Show that the equation 2x²− 6x +7 = 0 cannot be satisfied by any real values of x.

Show that if p, q, r  are rational the roots of the equation x² − 2px + p² − q² + 2qr − r² = 0 are rational.

Obtain the condition that the roots of x³+ px²+ qx + r = 0 are in A.P.

Discuss the nature of the roots of the following polynomials:
x²⁰¹⁸+1947x¹⁹⁵⁰+15x⁸+26x⁶+2019

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

Find the exact number of real zeros and imaginary of the polynomial x⁹+9x⁷+7x⁵+5x³+3x.

Construct a cubic equation with roots 2, −2, and 4.

Examine for the rational roots of x⁸- 3x + 1 = 0

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.

Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

Find the number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2