Construct a cubic equation with roots 1, 2 and 3

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

If p is real, discuss the nature of the roots of the equation 4x²+ 4px + p + 2 = 0 in terms of p.

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\(\delta\).

Find the monic polynomial equation of minimum degree with real coefficients having 2 -\(\sqrt{3}\)i as a root.

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

If x²+2(k+2)x+9k = 0 has equal roots, find k.

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

Find a polynomial equation of minimum degree with rational coefficients, having 2 +√3 i as a root.

Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

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

It is known that the roots of the equation x³- 6x²- 4x + 24 = 0 are in arithmetic progression. Find its roots.

Construct a cubic equation with roots 1, 1 and −2

If α, β and γ are the roots of the cubic equation x³+ 2x²+ 3x + 4 = 0, form a cubic equation whose roots are \(\frac { 1 }{ \alpha } ,\frac { 1 }{ \beta } ,\frac { 1 }{ \gamma } \)

If α, β and γ are the roots of the cubic equation x³+ 2x²+ 3x + 4 = 0, form a cubic equation whose roots are −α, -β, -γ

Construct a cubic equation with roots \(2, \frac{1}{2} \text { and } 1\)