Substitution to find new equation

4 Use the substitution \(x = u + 2\) to find the exact value of the real root of the equation $$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$

3 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 24 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence, or otherwise, find the value of \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\).

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

1. (a) Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha \beta + \beta \gamma + \gamma \alpha\).
   (b) Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. (a) Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
   (b) Use your answer to part (ii) (a) to find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).

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

1. Use the substitution \(x = u - 1\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).

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

1. Use the substitution \(x = 2 u + 1\) to obtain a quadratic equation in \(u\).
2. Hence, or otherwise, find the value of \(\left( \frac { \alpha - 1 } { 2 } \right) \left( \frac { \beta - 1 } { 2 } \right)\) in terms of \(k\).

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

1. Use the substitution \(x = \sqrt { u }\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).

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

1. Use the substitution \(x = \frac { 1 } { \sqrt { u } }\) to show that \(4 u ^ { 3 } + 12 u ^ { 2 } + 9 u - 1 = 0\).
2. Hence find the values of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) and \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).

6 The quadratic equation \(2 x ^ { 2 } + x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Use the substitution \(x = \frac { 1 } { u + 1 }\) to obtain a quadratic equation in \(u\) with integer coefficients.
2. Hence, or otherwise, find the value of \(\left( \frac { 1 } { \alpha } - 1 \right) \left( \frac { 1 } { \beta } - 1 \right)\).

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

1. Use the substitution \(x = u + 2\) to find a cubic equation in \(u\).
2. Hence find the value of \(\frac { 1 } { \alpha - 2 } + \frac { 1 } { \beta - 2 } + \frac { 1 } { \gamma - 2 }\).

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

1. Use the substitution \(x = \sqrt { u }\) to obtain a cubic equation in \(u\).
2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \alpha \beta \gamma\).

1 The equation $$x ^ { 4 } - x ^ { 3 } - 1 = 0$$ has roots \(\alpha , \beta , \gamma , \delta\). By using the substitution \(y = x ^ { 3 }\), or by any other method, find the exact value of \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } + \delta ^ { 6 }\).

6 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots \(\alpha , \beta , \gamma\). Use the relation \(x = \sqrt { } y\) to show that the equation $$y ^ { 3 } + 2 y ^ { 2 } + y - 1 = 0$$ has roots \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\). Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).

1. Write down the value of \(S _ { 2 }\) and show that \(S _ { 4 } = 2\).
2. Find the values of \(S _ { 6 }\) and \(S _ { 8 }\).

10 The equation $$x ^ { 4 } + x ^ { 3 } + c x ^ { 2 } + 4 x - 2 = 0$$ where \(c\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\).

1. Use the substitution \(y = \frac { 1 } { x }\) to find an equation which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }\).
2. Find, in terms of \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\) and \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
3. Hence find $$\left( \alpha - \frac { 1 } { \alpha } \right) ^ { 2 } + \left( \beta - \frac { 1 } { \beta } \right) ^ { 2 } + \left( \gamma - \frac { 1 } { \gamma } \right) ^ { 2 } + \left( \delta - \frac { 1 } { \delta } \right) ^ { 2 }$$ in terms of \(c\).
4. Deduce that when \(c = - 3\) the roots of the given equation are not all real.

3 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 3 x + 4 = 0\) has roots \(\alpha , \beta , \gamma\). Given that \(c = \alpha + \beta + \gamma\), state the value of \(c\). Use the substitution \(y = c - x\) to find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\). Find a cubic equation whose roots are \(\frac { 1 } { \alpha + \beta } , \frac { 1 } { \beta + \gamma } , \frac { 1 } { \gamma + \alpha }\). Hence evaluate \(\frac { 1 } { ( \alpha + \beta ) ^ { 2 } } + \frac { 1 } { ( \beta + \gamma ) ^ { 2 } } + \frac { 1 } { ( \gamma + \alpha ) ^ { 2 } }\).

2 The roots of the equation \(x ^ { 4 } - 4 x ^ { 2 } + 3 x - 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\); the sum \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). By using the relation \(y = x ^ { 2 }\), or otherwise, show that \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\) and \(\delta ^ { 2 }\) are the roots of the equation $$y ^ { 4 } - 8 y ^ { 3 } + 12 y ^ { 2 } + 7 y + 4 = 0$$ State the value of \(S _ { 2 }\) and hence show that $$S _ { 8 } = 8 S _ { 6 } - 12 S _ { 4 } - 72 .$$

1 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 7 = 0\) are \(\alpha , \beta\) and \(\gamma\). Using the substitution \(y = 1 + \frac { 1 } { x }\), or otherwise, find the cubic equation whose roots are \(1 + \frac { 1 } { \alpha } , 1 + \frac { 1 } { \beta }\) and \(1 + \frac { 1 } { \gamma }\), giving your answer in the form \(a y ^ { 3 } + b y ^ { 2 } + c y + d = 0\), where \(a , b , c\) and \(d\) are constants to be found.

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

1. By using the substitution \(y = \frac { 1 } { x ^ { 2 } }\), find the cubic equation with roots \(\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } }\) and \(\frac { 1 } { \gamma ^ { 2 } }\).
2. Hence find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\).
3. Find also the value of \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).

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

1. Use the substitution \(y = 3 x - 1\) to show that \(3 \alpha - 1,3 \beta - 1,3 \gamma - 1\) are the roots of the equation $$y ^ { 3 } - 2 y - 7 = 0$$ The sum \(( 3 \alpha - 1 ) ^ { n } + ( 3 \beta - 1 ) ^ { n } + ( 3 \gamma - 1 ) ^ { n }\) is denoted by \(S _ { n }\).
2. Find the value of \(S _ { 3 }\).
3. Find the value of \(S _ { - 2 }\).

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

1. Use the relation \(x = y ^ { \frac { 1 } { 3 } }\) to show that the equation $$y ^ { 3 } + 3 y ^ { 2 } + 2 y + 1 = 0$$ has roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\). Hence write down the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
   Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).
2. Find the value of \(S _ { - 3 }\).
3. Show that \(S _ { 6 } = 5\) and find the value of \(S _ { 9 }\).

1 The roots of the equation \(4 x ^ { 4 } - 2 x ^ { 3 } - 3 x + 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). By using a suitable substitution, find a quartic equation whose roots are \(\alpha + 2 , \beta + 2 , \gamma + 2\) and \(\delta + 2\) giving your answer in the form \(a t ^ { 4 } + b t ^ { 3 } + c t ^ { 2 } + d t + e = 0\), where \(a , b , c , d\), and \(e\) are integers.

2 The equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Use a substitution to find a cubic equation with integer coefficients whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).

1. The cubic equation

$$x ^ { 3 } + 3 x ^ { 2 } - 8 x + 6 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \(( \alpha - 1 ) , ( \beta - 1 )\) and \(( \gamma - 1 )\), 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.
(5)