Equation with linearly transformed roots

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 \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).

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 \(2 \beta\).

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.

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.

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.

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 \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\).

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 \(\frac { 1 } { \gamma }\).

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 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 \(2 \gamma - 1\).

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.

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. The quartic equation $$z ^ { 4 } + 5 z ^ { 2 } - 30 = 0$$ has roots \(p , q , r\) and \(s\).
   Without solving the equation, determine the quartic equation whose roots are $$( 3 p - 1 ) , ( 3 q - 1 ) , ( 3 r - 1 ) \text { and } ( 3 s - 1 )$$ Give your answer in the form \(w ^ { 4 } + a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be found.
2. The roots of the cubic equation $$4 x ^ { 3 } + n x + 81 = 0 \quad \text { where } n \text { is a real constant }$$ are \(\alpha , 2 \alpha\) and \(\alpha - \beta\) Determine
   (a) the values of the roots of the equation,
   (b) the value of \(n\).

1. The cubic equation

$$x ^ { 3 } - 7 x ^ { 2 } - 12 x + 6 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, determine a cubic equation whose roots are ( \(\alpha + 2\) ), \(( \beta + 2 )\) and \(( \gamma + 2 )\), 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.

4 In this question you must show detailed reasoning. The roots of the cubic equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { 2 } ( \alpha - 1 ) , \frac { 1 } { 2 } ( \beta - 1 )\) and \(\frac { 1 } { 2 } ( \gamma - 1 )\).
2. Hence or otherwise solve the equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).

7 The cubic equation \(27 z ^ { 3 } + k z ^ { 2 } + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. 1. In the case where \(\beta = \gamma\), find the roots of the equation.
   2. Find the value of \(k\) in this case.
3. 1. In the case where \(\alpha = 1 - \mathrm { i }\), find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\).
   2. Hence find the value of \(k\) in this case.
4. In the case where \(k = - 12\), find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1\) and \(\frac { 1 } { \gamma } + 1\).
   [0pt] [5 marks]

13 The cubic equation \(x ^ { 3 } - x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\) The cubic equation \(\mathrm { p } ( x ) = 0\) has roots \(\alpha - 1 , \beta - 1\) and \(\gamma - 1\) The coefficient of \(x ^ { 3 }\) in \(\mathrm { p } ( x )\) is 1 13

1. Describe fully the transformation that maps the graph of \(y = x ^ { 3 } - x - 7\) onto the graph of \(y = \mathrm { p } ( x )\) 13
2. Find \(\mathrm { p } ( x )\) Turn over for the next question