Equation with transformed roots

5 The roots of the cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha - 1,3 \beta - 1\) and \(3 \gamma - 1\), expressing your answer in a form with integer coefficients.

5 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots \(\alpha , \beta , \gamma\). Show that the equation with roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\) is $$y ^ { 3 } - 3 y ^ { 2 } + 4 y - 1 = 0$$ Hence find the value of \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 }\).

8 The cubic equation \(x ^ { 3 } - x ^ { 2 } - 3 x - 10 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Let \(u = - \alpha + \beta + \gamma\). Show that \(u + 2 \alpha = 1\), and hence find a cubic equation having roots \(- \alpha + \beta + \gamma\), \(\alpha - \beta + \gamma , \alpha + \beta - \gamma\).
2. State the value of \(\alpha \beta \gamma\) and hence find a cubic equation having roots \(\frac { 1 } { \beta \gamma } , \frac { 1 } { \gamma \alpha } , \frac { 1 } { \alpha \beta }\).

4 The roots of the cubic equation \(x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the values of

1. \(\frac { 1 } { ( \alpha \beta ) ( \beta \gamma ) ( \gamma \alpha ) }\),
2. \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\),
3. \(\frac { 1 } { \alpha ^ { 2 } \beta \gamma } + \frac { 1 } { \alpha \beta ^ { 2 } \gamma } + \frac { 1 } { \alpha \beta \gamma ^ { 2 } }\). Deduce a cubic equation, with integer coefficients, having roots \(\frac { 1 } { \alpha \beta } , \frac { 1 } { \beta \gamma }\) and \(\frac { 1 } { \gamma \alpha }\).

2 The equation $$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$ where \(A\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\). Find a polynomial equation whose roots are $$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$ Given that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$ find the value of \(A\).

6 The roots of the equation $$x ^ { 3 } + x + 1 = 0$$ are \(\alpha , \beta , \gamma\). Show that the equation whose roots are $$\frac { 4 \alpha + 1 } { \alpha + 1 } , \quad \frac { 4 \beta + 1 } { \beta + 1 } , \quad \frac { 4 \gamma + 1 } { \gamma + 1 }$$ is of the form $$y ^ { 3 } + p y + q = 0$$ where the numbers \(p\) and \(q\) are to be determined. Hence find the value of $$\left( \frac { 4 \alpha + 1 } { \alpha + 1 } \right) ^ { n } + \left( \frac { 4 \beta + 1 } { \beta + 1 } \right) ^ { n } + \left( \frac { 4 \gamma + 1 } { \gamma + 1 } \right) ^ { n }$$ for \(n = 2\) and for \(n = 3\).

5 The equation $$x ^ { 3 } + 5 x + 3 = 0$$ has roots \(\alpha , \beta , \gamma\). Use the substitution \(x = - \frac { 3 } { y }\) to find a cubic equation in \(y\) and show that the roots of this equation are \(\beta \gamma , \gamma \alpha , \alpha \beta\). Find the exact values of \(\beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 }\) and \(\beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } + \alpha ^ { 3 } \beta ^ { 3 }\).

7 The roots of the equation \(x ^ { 3 } + 4 x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = \frac { 1 } { 1 + x }\) to show that the equation \(6 y ^ { 3 } - 7 y ^ { 2 } + 3 y - 1 = 0\) has roots \(\frac { 1 } { \alpha + 1 } , \frac { 1 } { \beta + 1 }\) and \(\frac { 1 } { \gamma + 1 }\). For the cases \(n = 1\) and \(n = 2\), find the value of $$\frac { 1 } { ( \alpha + 1 ) ^ { n } } + \frac { 1 } { ( \beta + 1 ) ^ { n } } + \frac { 1 } { ( \gamma + 1 ) ^ { n } }$$ Deduce the value of \(\frac { 1 } { ( \alpha + 1 ) ^ { 3 } } + \frac { 1 } { ( \beta + 1 ) ^ { 3 } } + \frac { 1 } { ( \gamma + 1 ) ^ { 3 } }\). Hence show that \(\frac { ( \beta + 1 ) ( \gamma + 1 ) } { ( \alpha + 1 ) ^ { 2 } } + \frac { ( \gamma + 1 ) ( \alpha + 1 ) } { ( \beta + 1 ) ^ { 2 } } + \frac { ( \alpha + 1 ) ( \beta + 1 ) } { ( \gamma + 1 ) ^ { 2 } } = \frac { 73 } { 36 }\).

The roots of the quartic equation \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). Find the values of

1. \(\alpha + \beta + \gamma + \delta\),
2. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\),
3. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } + \frac { 1 } { \delta }\),
4. \(\frac { \alpha } { \beta \gamma \delta } + \frac { \beta } { \alpha \gamma \delta } + \frac { \gamma } { \alpha \beta \delta } + \frac { \delta } { \alpha \beta \gamma }\). Using the substitution \(y = x + 1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\).

7 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 7 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Use the relation \(x ^ { 2 } = - 7 y\) to show that the equation $$49 y ^ { 3 } + 14 y ^ { 2 } - 27 y + 7 = 0$$ has roots \(\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }\).
2. Show that \(\frac { \alpha ^ { 2 } } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { \beta ^ { 2 } } { \gamma ^ { 2 } \alpha ^ { 2 } } + \frac { \gamma ^ { 2 } } { \alpha ^ { 2 } \beta ^ { 2 } } = \frac { 58 } { 49 }\).
3. Find the exact value of \(\frac { \alpha ^ { 3 } } { \beta ^ { 3 } \gamma ^ { 3 } } + \frac { \beta ^ { 3 } } { \gamma ^ { 3 } \alpha ^ { 3 } } + \frac { \gamma ^ { 3 } } { \alpha ^ { 3 } \beta ^ { 3 } }\).

2 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

7 In this question you must show detailed reasoning.
The roots of the equation \(2 x ^ { 3 } - 3 x ^ { 2 } - 3 x + 5 = 0\) are \(\alpha , \beta\) and \(\gamma\).
By considering \(( \alpha + \beta + \gamma ) ^ { 2 }\) and \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 }\), determine a cubic equation with integer coefficients whose roots are \(\frac { \alpha \beta } { \gamma } , \frac { \beta \gamma } { \alpha }\) and \(\frac { \gamma \alpha } { \beta }\).

2 In this question you must show detailed reasoning.
The equation \(z ^ { 4 } + 4 z ^ { 3 } + 9 z ^ { 2 } + 10 z + 6 = 0\) has roots \(\alpha , \beta , \gamma\) and \(\delta\).

1. Show that a quartic equation whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\) and \(\delta + 1\) is \(w ^ { 4 } + 3 w ^ { 2 } + 2 = 0\).
2. Hence determine the exact roots of the equation \(z ^ { 4 } + 4 z ^ { 3 } + 9 z ^ { 2 } + 10 z + 6 = 0\).

4 In this question you must show detailed reasoning.
The equation \(2 x ^ { 3 } + 3 x ^ { 2 } + 6 x - 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Determine a cubic equation with integer coefficients that has roots \(\alpha ^ { 2 } \beta \gamma , \alpha \beta ^ { 2 } \gamma\) and \(\alpha \beta \gamma ^ { 2 }\).

3 A function \(\mathrm { f } ( \mathrm { z } )\) is defined on all complex numbers z by \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 3 \mathrm { z } ^ { 2 } + \mathrm { k } \mathrm { z } - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta\) and \(\gamma\). You are given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5\).

1. Explain why \(f ( z ) = 0\) has only one real root.
2. Find the value of \(k\).
3. Find a cubic equation with integer coefficients that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

6 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
Find the values of \(p , q\) and \(r\).

2 In this question you must show detailed reasoning.
The roots of the equation \(3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find a cubic equation with integer coefficients whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
2. Find the exact value of \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).

4 The roots of the equation $$z ^ { 3 } + 2 z ^ { 2 } + 3 z - 4 = 0$$ are \(\alpha , \beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
   2. Hence show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 2\).
2. Find the value of:
   1. \(( \alpha + \beta ) ( \beta + \gamma ) + ( \beta + \gamma ) ( \gamma + \alpha ) + ( \gamma + \alpha ) ( \alpha + \beta )\);
   2. \(( \alpha + \beta ) ( \beta + \gamma ) ( \gamma + \alpha )\).
3. Find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

4 The roots of the equation \(2 x ^ { 3 } - 5 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
2. Find an equation with integer coefficients whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).

10. The quadratic equation \(p x ^ { 2 } + q x + r = 0\) has roots \(\alpha\) and \(\beta\), where \(p , q , r\) are non-zero constants.

1. A cubic equation is formed with roots \(\alpha , \beta , \alpha + \beta\). Find the cubic equation with coefficients expressed in terms of \(p , q , r\).
2. Another quadratic equation \(p x ^ { 2 } - q x - r = 0\) has roots \(2 \alpha\) and \(\gamma\). Show that \(\beta = - 2 \gamma\).

8. The roots of the cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 2 x + 8 = 0\) are denoted by \(\alpha , \beta , \gamma\). Determine the cubic equation whose roots are \(\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }\).
Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), where \(a , b , c , d\) are constants to be determined.

1. The cubic equation

$$z ^ { 3 } - 3 z ^ { 2 } + z + 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(2 \alpha + 1\) ), ( \(2 \beta + 1\) ) and ( \(2 \gamma + 1\) ), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.

1. The cubic equation

$$2 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 12 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \(( \alpha + 3 ) , ( \beta + 3 )\) and \(( \gamma + 3 )\), giving your answer in the form \(p w ^ { 3 } + q w ^ { 2 } + r w + s = 0\), where \(p , q , r\) and \(s\) are integers to be found.

1. The cubic equation

$$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha , \beta\), and \(\gamma\).
Without solving the cubic equation,

1. determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
2. find a cubic equation that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\), giving your answer in the form \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers to be determined.

1. The cubic equation

$$9 x ^ { 3 } - 5 x ^ { 2 } + 4 x + 7 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(3 \alpha - 2\) ), ( \(3 \beta - 2\) ) and ( \(3 \gamma - 2\) ), giving your answer in the form \(a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be determined.