4.05b Transform equations: substitution for new roots

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 \(\frac { \alpha ^ { 3 } } { \beta ^ { 3 } \gamma ^ { 3 } } + \frac { \beta ^ { 3 } } { \gamma ^ { 3 } \alpha ^ { 3 } } + \frac { \gamma ^ { 3 } } { \alpha ^ { 3 } \beta ^ { 3 } }\).

1 The quartic equation \(x ^ { 4 } - p x ^ { 2 } + q x - r = 0\), where \(p , q\) and \(r\) are real constants, has two pairs of equal roots. Show that \(p ^ { 2 } + 4 r = 0\) and state the value of \(q\).

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 \(\frac { \gamma \alpha } { \beta }\).

1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\).

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 \(\alpha \beta \gamma ^ { 2 }\).

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 \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = - \frac { 9 } { 5 }\).
4. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).

1 The equation $$2 x ^ { 2 } + x - 8 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(4 \alpha ^ { 2 }\) and \(4 \beta ^ { 2 }\). Give your answer in the form \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are integers.

8 The quadratic equation $$x ^ { 2 } - 4 x + 10 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 2 } { 5 }\).
3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha + \frac { 2 } { \beta }\) and \(\beta + \frac { 2 } { \alpha }\).

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = \frac { 3 } { 4 }\).
3. Find an equation, with integer coefficients, which has roots $$3 \alpha - \beta \text { and } 3 \beta - \alpha$$

1 The quadratic equation $$5 x ^ { 2 } - 7 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
3. Find a quadratic equation, with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$ (5 marks)

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

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Hence show that \(\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }\).
3. Find a quadratic equation, with integer coefficients, whose roots are \(\alpha + \frac { \alpha } { \beta ^ { 2 } }\) and \(\beta + \frac { \beta } { \alpha ^ { 2 } }\).

1 The quadratic equation \(2 x ^ { 2 } + 6 x + 7 = 0\) has roots \(\alpha\) and \(\beta\).

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

8

1. Write down the five roots of the equation \(z ^ { 5 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Hence find the four linear factors of $$z ^ { 4 } + z ^ { 3 } + z ^ { 2 } + z + 1$$
3. Deduce that $$z ^ { 2 } + z + 1 + z ^ { - 1 } + z ^ { - 2 } = \left( z - 2 \cos \frac { 2 \pi } { 5 } + z ^ { - 1 } \right) \left( z - 2 \cos \frac { 4 \pi } { 5 } + z ^ { - 1 } \right)$$
4. Use the substitution \(z + z ^ { - 1 } = w\) to show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).

2 The equation \(3 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\alpha\) and \(\beta\).
Find an equation with integer coefficients whose roots are \(3 - 2 \alpha\) and \(3 - 2 \beta\).

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 \(2 \gamma - 1\).

5. Given that \(x = - \frac { 1 } { 2 }\) and \(x = - 3\) are two roots of the equation $$2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 23 x + 15 = 0$$ find the remaining roots.

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 \(\beta = - 2 \gamma\).

6. The roots of the cubic equation $$2 x ^ { 3 } + p x ^ { 2 } - 126 x + q = 0$$ form a geometric progression with common ratio - 3 .
Find the possible values of \(p\) and \(q\), giving your answers in surd form.

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.

3. The quadratic equation \(x ^ { 2 } + p x + q = 0\) has a repeated root \(\alpha\). A new quadratic equation has a repeated root \(\frac { 1 } { \alpha }\) and is of the form \(x ^ { 2 } + m x + m = 0\).
Find the values of \(p\) and \(q\) in the original equation.
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