Equation with transformed roots

2 The cubic equation \(6 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } - 3 \mathrm { x } - 5 = 0\), where \(p\) is a constant, has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\).
2. It is given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 ( \alpha + \beta + \gamma )\).
   1. Find the value of \(p\).
   2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

1 The cubic equation \(7 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { - 1 } , \beta ^ { - 1 } , \gamma ^ { - 1 }\).
2. Find the value of \(\alpha ^ { - 2 } + \beta ^ { - 2 } + \gamma ^ { - 2 }\).
3. Find the value of \(\alpha ^ { - 3 } + \beta ^ { - 3 } + \gamma ^ { - 3 }\).

3 The equation \(x ^ { 4 } - 2 x ^ { 3 } - 1 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).

1. Find a quartic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 } , \delta ^ { 3 }\) and state the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 }\). [4]
2. Find the value of \(\frac { 1 } { \alpha ^ { 3 } } + \frac { 1 } { \beta ^ { 3 } } + \frac { 1 } { \gamma ^ { 3 } } + \frac { 1 } { \delta ^ { 3 } }\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

4 The cubic equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 6 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\frac { 1 } { \alpha ^ { 3 } } , \frac { 1 } { \beta ^ { 3 } } , \frac { 1 } { \gamma ^ { 3 } }\).
2. Find the value of \(\frac { 1 } { \alpha ^ { 6 } } + \frac { 1 } { \beta ^ { 6 } } + \frac { 1 } { \gamma ^ { 6 } }\).
3. Find also the value of \(\frac { 1 } { \alpha ^ { 9 } } + \frac { 1 } { \beta ^ { 9 } } + \frac { 1 } { \gamma ^ { 9 } }\).

3 The equation \(x ^ { 4 } - x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).

1. Find a quartic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 } , \delta ^ { 2 }\) and state the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\).
2. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

2 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 2 } + 1 , \beta ^ { 2 } + 1 , \gamma ^ { 2 } + 1\).
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2. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 2 } + \left( \beta ^ { 2 } + 1 \right) ^ { 2 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 2 }\).
3. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 3 } + \left( \beta ^ { 2 } + 1 \right) ^ { 3 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 3 }\).

3 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { cx } + 1 = 0\), where \(c\) is a constant, has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\).
2. Show that \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } = 3 - 2 c ^ { 3 }\).
3. Find the real value of \(c\) for which the matrix \(\left( \begin{array} { c c c } 1 & \alpha ^ { 3 } & \beta ^ { 3 } \\ \alpha ^ { 3 } & 1 & \gamma ^ { 3 } \\ \beta ^ { 3 } & \gamma ^ { 3 } & 1 \end{array} \right)\) is singular.

1 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\), where \(b , c\) and \(d\) are constants, has roots \(\alpha , \beta , \gamma\). It is given that \(\alpha \beta \gamma = - 1\).

1. State the value of \(d\).
2. Find a cubic equation, with coefficients in terms of \(b\) and \(c\), whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\).
3. Given also that \(\gamma + 1 = - \alpha - 1\), deduce that \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).

2 The equation \(x ^ { 4 } + 3 x ^ { 2 } + 2 x + 6 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).

1. Find a quartic equation whose roots are \(\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } } , \frac { 1 } { \gamma ^ { 2 } } , \frac { 1 } { \delta ^ { 2 } }\) and state the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
2. Find the value of \(\beta ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
3. Find the value of \(\frac { 1 } { \alpha ^ { 4 } } + \frac { 1 } { \beta ^ { 4 } } + \frac { 1 } { \gamma ^ { 4 } } + \frac { 1 } { \delta ^ { 4 } }\).

4 The cubic equation \(27 x ^ { 3 } + 18 x ^ { 2 } + 6 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Show that a cubic equation with roots \(3 \alpha + 1,3 \beta + 1,3 \gamma + 1\) is $$y ^ { 3 } - y ^ { 2 } + y - 2 = 0$$ The sum \(( 3 \alpha + 1 ) ^ { n } + ( 3 \beta + 1 ) ^ { n } + ( 3 \gamma + 1 ) ^ { n }\) is denoted by \(\mathrm { S } _ { \mathrm { n } }\).
2. Find the values of \(S _ { 2 }\) and \(S _ { 3 }\).
3. Find the values of \(S _ { - 1 }\) and \(S _ { - 2 }\).

3 The quartic equation \(x ^ { 4 } + 2 x ^ { 3 } - 1 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).

1. Find a quartic equation whose roots are \(\alpha ^ { 4 } , \beta ^ { 4 } , \gamma ^ { 4 } , \delta ^ { 4 }\) and state the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
2. Find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }\).
3. Find the value of \(\alpha ^ { 8 } + \beta ^ { 8 } + \gamma ^ { 8 } + \delta ^ { 8 }\).

9

1. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )\).
2. The quadratic equation \(x ^ { 2 } - 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\).
3. Show that \(\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
4. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
5. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
6. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }\), find the value of \(N\).

8

1. The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
   1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
   2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 4\).
   3. Hence find a quadratic equation which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
2. The cubic equation \(x ^ { 3 } - 12 x ^ { 2 } + a x - 48 = 0\) has roots \(p , 2 p\) and \(3 p\).
   1. Find the value of \(p\).
   2. Hence find the value of \(a\).

10 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 3 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
2. Find the value of \(p\).
3. Find the value of \(q\).

5 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), simplifying your answer as far as you can.

5 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Find the cubic equation with roots \(2 \alpha , 2 \beta\) and \(2 \gamma\), simplifying your answer as far as possible.

5 The roots of the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(2 \alpha + 1,2 \beta + 1\) and \(2 \gamma + 1\), expressing your answer in a form with integer coefficients.

4 The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(2 \alpha\) and \(2 \beta\).

5 The roots of the cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha , 3 \beta\) and \(3 \gamma\), expressing your answer in a form with integer coefficients.

1. Show that \(\frac { 1 } { r + 2 } - \frac { 1 } { r + 3 } = \frac { 1 } { ( r + 2 ) ( r + 3 ) }\).
2. Hence use the method of differences to find \(\frac { 1 } { 3 \times 4 } + \frac { 1 } { 4 \times 5 } + \frac { 1 } { 5 \times 6 } + \ldots + \frac { 1 } { 52 \times 53 }\).

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

10 The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\) has roots \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
2. Show that \(c = - \frac { 4 } { 9 }\) and find the values of \(a\) and \(b\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

3 The roots of the cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).

5 The equation \(z ^ { 3 } - 5 z ^ { 2 } + 3 z - 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(\frac { \alpha } { 2 } + 1 , \frac { \beta } { 2 } + 1\), \(\frac { \gamma } { 2 } + 1\), expressing your answer in a form with integer coefficients.

4 The roots of the cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 8 x + 11 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).

6 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + 3 x - 6 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with roots \(\frac { \alpha } { 3 } + 1 , \frac { \beta } { 3 } + 1\) and \(\frac { \gamma } { 3 } + 1\), simplifying your answer as far as possible.