4.05b Transform equations: substitution for new roots

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

3 In this question you must show detailed reasoning.
The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

1 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 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)

1 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = 1 + x ^ { 2 }\) to find an equation, with integer coefficients, whose roots are \(1 + \alpha ^ { 2 } , 1 + \beta ^ { 2 }\) and \(1 + \gamma ^ { 2 }\).

8 The equation \(8 x ^ { 3 } + 12 x ^ { 2 } + 4 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. By considering a suitable substitution, or otherwise, show that the equation whose roots are \(2 \alpha + 1,2 \beta + 1,2 \gamma + 1\) can be written in the form $$y ^ { 3 } - y - 1 = 0 .$$
2. The sum \(( 2 \alpha + 1 ) ^ { n } + ( 2 \beta + 1 ) ^ { n } + ( 2 \gamma + 1 ) ^ { n }\) is denoted by \(S _ { n }\). Evaluate \(S _ { 3 }\) and \(S _ { - 2 }\).

The quartic equation \(x^4 + 2x^3 - 1 = 0\) 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 \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\). [5]
2. Find the value of \(\alpha^5 + \beta^5 + \gamma^5 + \delta^5\). [3]
3. Find the value of \(\alpha^8 + \beta^8 + \gamma^8 + \delta^8\). [2]

The cubic equation \(x^3 + px^2 + qx + r = 0\), where \(p\), \(q\) and \(r\) are integers, has roots \(\alpha\), \(\beta\) and \(\gamma\), such that $$\alpha + \beta + \gamma = 15,$$ $$\alpha^2 + \beta^2 + \gamma^2 = 83.$$ Write down the value of \(p\) and find the value of \(q\). [3] Given that \(\alpha\), \(\beta\) and \(\gamma\) are all real and that \(\alpha\beta + \alpha\gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]

Using de Moivre's theorem, show that $$\tan 5\theta = \frac{5\tan\theta - 10\tan^3\theta + \tan^5\theta}{1 - 10\tan^2\theta + 5\tan^4\theta}.$$ [5] Hence show that the equation \(x^2 - 10x + 5 = 0\) has roots \(\tan^2\left(\frac{1}{5}\pi\right)\) and \(\tan^2\left(\frac{2}{5}\pi\right)\). [4] Deduce a quadratic equation, with integer coefficients, having roots \(\sec^2\left(\frac{1}{5}\pi\right)\) and \(\sec^2\left(\frac{2}{5}\pi\right)\). [3]

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

1. Use the relation \(x^2 = -7y\) to show that the equation $$49y^3 + 14y^2 - 27y + 7 = 0$$ has roots \(\frac{\alpha}{\beta \gamma}\), \(\frac{\beta}{\gamma \alpha}\), \(\frac{\gamma}{\alpha \beta}\). [4]
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]
3. Find the exact value of \(\frac{\alpha^2}{\beta^3 \gamma^3} + \frac{\beta^2}{\gamma^3 \alpha^3} + \frac{\gamma^2}{\alpha^3 \beta^3}\). [2]

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

1. Use the relation \(x^2 = -7y\) to show that the equation $$49y^3 + 14y^2 - 27y + 7 = 0$$ has roots \(\frac{\alpha}{\beta\gamma}\), \(\frac{\beta}{\gamma\alpha}\), \(\frac{\gamma}{\alpha\beta}\). [4]
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]
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]

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

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
2. 1. Find the value of \(\alpha^2 + \beta^2\). [2 marks]
   2. Hence, or otherwise, show that \(\alpha^4 + \beta^4 = \frac{449}{2}\). [2 marks]
3. Find a quadratic equation, with integer coefficients, which has roots $$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$ [5 marks]

The quadratic equation \(x^2 - 6x + 14 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
2. Find a quadratic equation, with integer coefficients, which has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). [5 marks]

The roots of the cubic equation \(x^3 + 3x^2 - 7x + 1 = 0\) 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. [6]

The quadratic equation \(2x^2 + px + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\).

1. 1. Write down the value of \(\alpha\beta\). [1 mark]
   2. Express \(\alpha + \beta\) in terms of \(p\). [1 mark]
2. Hence find \((\alpha - \beta)^2\) in terms of \(p\). [2 marks]
3. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\) [4 marks]

The cubic equation $$x^3 + 5x^2 - 4x + 2 = 0$$ has roots \(\alpha\), \(\beta\) and \(\gamma\) Find a cubic equation, with integer coefficients, whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\) [3 marks]

The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]

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

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

A cubic equation has roots \(\alpha\), \(\beta\), \(\gamma\) such that $$\alpha + \beta + \gamma = -9, \quad \alpha\beta + \beta\gamma + \gamma\alpha = 20, \quad \alpha\beta\gamma = 0.$$

1. Find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
2. Find the cubic equation with roots \(3\alpha\), \(3\beta\), \(3\gamma\). Give your answer in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), \(d\) are constants to be determined. [4]

The equation \(4x^4 - 4x^3 + px^2 + qx - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha\), \(-\alpha\), \(\beta\) and \(\frac{1}{\beta}\).

1. Determine the exact roots of the equation. [5]
2. Determine the values of \(p\) and \(q\). [4]

The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]

In this question you must show detailed reasoning. The roots of the equation \(2x^3 - 5x + 7 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\).

1. Find \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]
2. Find an equation with integer coefficients whose roots are \(2\alpha - 1\), \(2\beta - 1\) and \(2\gamma - 1\). [4]

The cubic equation $$2x^3 + 6x^2 - 3x + 12 = 0$$ 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 \(pw^3 + qw^2 + rw + s = 0\), where \(p\), \(q\), \(r\) and \(s\) are integers to be found. [5]