Roots of polynomials

The equation 5x³ - 4x² + 10 = 0 has roots α, β and γ.

Write down the values of α + β + γ, αβ + βγ + γα and αβγ.

By expanding (αβ + βγ + γα)², determine the value of α²β² + β²γ² + γ²α².

By expanding a suitable expression, find the value of α² + β² + γ².

Hence find a cubic equation with integer coefficients that has roots α², β², and γ².

In this question you must show detailed reasoning.

The equation 2x³ + 3x² + 6x - 3 = 0 has roots α, β and γ.

Determine a cubic equation with integer coefficients that has roots α²βγ, αβ²γ and αβγ².

In this question you must show detailed reasoning.

The equation z⁴ + 4z³ + 9z² + 10z + 6 = 0 has roots α, β, γ and δ.

Show that a quartic equation whose roots are α + 1, β + 1, γ + 1 and δ + 1 is

w⁴ + 3w² + 2 = 0.

Hence determine the exact roots of the equation z⁴ + 4z³ + 9z² + 10z + 6 = 0.

A function f(z) is defined on all complex numbers z by f(z) = z³ - 3z² + kz - 5 where k is a real constant. The roots of the equation f(z) = 0 are α, β and γ. You are given that α² + β² + γ² =-5.

Explain why f (z) = 0 has only one real root.

Find the value of k.

You are given that the cubic equation 2x³ + px² + qx - 3 = 0, where p and q are real numbers, has a

Write down a second complex root, β.

Determine the third root, γ.

Find the value of p and the value of q.