Finding polynomial from root properties

1 It is given that $$\alpha + \beta + \gamma = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 5 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 6 .$$ The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\) has roots \(\alpha , \beta , \gamma\).
Find the values of \(b , c\) and \(d\).

3 The quartic equation \(\mathrm { x } ^ { 4 } + \mathrm { bx } ^ { 3 } + \mathrm { cx } ^ { 2 } + \mathrm { dx } - 2 = 0\) has roots \(\alpha , \beta , \gamma , \delta\). It is given that $$\alpha + \beta + \gamma + \delta = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 5 , \quad \alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { - 1 } + \delta ^ { - 1 } = 6$$

1. Find the values of \(b , c\) and \(d\).
2. Given also that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = - 27\), find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

5 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha \beta \gamma & = - 7 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 13 \end{aligned}$$

1. Write down the values of \(p\) and \(r\).
2. Find the value of \(q\).

3 Find a cubic equation with roots \(\alpha , \beta\) and \(\gamma\), given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.

7 By finding a cubic equation whose roots are \(\alpha , \beta\) and \(\gamma\), solve the set of simultaneous equations $$\begin{aligned} \alpha + \beta + \gamma & = - 1 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 29 , \\ \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - 1 . \end{aligned}$$

9 A cubic equation \(x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) has real roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - \frac { 5 } { 12 } \\ \alpha \beta \gamma & = - 12 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 90 \end{aligned}$$

1. Find the values of \(c\) and \(d\).
2. Express \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) in terms of \(b\).
3. Show that \(b ^ { 3 } - 15 b + 126 = 0\).
4. Given that \(3 + \mathrm { i } \sqrt { } ( 12 )\) is a root of \(y ^ { 3 } - 15 y + 126 = 0\), deduce the value of \(b\).

3 Given that $$\alpha + \beta + \gamma = 0 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 14 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 18$$ find a cubic equation whose roots are \(\alpha , \beta , \gamma\). Hence find possible values for \(\alpha , \beta , \gamma\).

2 Find the cubic equation with roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30 \end{aligned}$$ giving your answer in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers to be found.

7 A cubic equation has roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 4 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 14 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 34 \end{aligned}$$ Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\). Show that the cubic equation is $$x ^ { 3 } - 4 x ^ { 2 } + x + 6 = 0$$ and solve this equation.

4 It is given that \(\alpha , \beta\) and \(\gamma\) satisfy the equations $$\begin{aligned} & \alpha + \beta + \gamma = 1 \\ & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5 \\ & \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 23 \end{aligned}$$

1. Show that \(\alpha \beta + \beta \gamma + \gamma \alpha = 3\).
2. Use the identity $$( \alpha + \beta + \gamma ) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } - \alpha \beta - \beta \gamma - \gamma \alpha \right) = \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } - 3 \alpha \beta \gamma$$ to find the value of \(\alpha \beta \gamma\).
3. Write down a cubic equation, with integer coefficients, whose roots are \(\alpha , \beta\) and \(\gamma\).
4. Explain why this cubic equation has two non-real roots.
5. Given that \(\alpha\) is real, find the values of \(\alpha , \beta\) and \(\gamma\).

5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 . \end{aligned}$$ Write down the value of \(p\) and find the value of \(q\). Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).

In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
2. Given that \(\alpha^3\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^3\), \(\beta^3\) and \(\gamma^3\). [4]