4.05a Roots and coefficients: symmetric functions

The roots of the cubic equation $$x^3 - 5x^2 + 13x - 4 = 0$$ are \(\alpha, \beta, \gamma\).

1. Find the value of \(\alpha^2 + \beta^2 + \gamma^2\). [3]
2. Find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [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 \(\frac{\alpha^2}{\beta^3 \gamma^3} + \frac{\beta^2}{\gamma^3 \alpha^3} + \frac{\gamma^2}{\alpha^3 \beta^3}\). [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 equation $$x^4 + Ax^3 + Bx^2 + Cx + 225 = 0$$ where \(A\), \(B\) and \(C\) are real constants, has

• a complex root \(4 + 3\text{i}\)
• a repeated positive real root

1. Write down the other complex root of this equation. [1]
2. Hence determine a quadratic factor of \(x^4 + Ax^3 + Bx^2 + Cx + 225\) [2]
3. Deduce the real root of the equation. [2]
4. Hence determine the value of each of the constants \(A\), \(B\) and \(C\) [3]

The quadratic equation $$Ax^2 + 5x - 12 = 0$$ where \(A\) is a constant, has roots \(\alpha\) and \(\beta\)

1. Write down an expression in terms of \(A\) for
   1. \(\alpha + \beta\)
   2. \(\alpha\beta\)
   [2]

The equation $$4x^2 - 5x + B = 0$$ where \(B\) is a constant, has roots \(\alpha - \frac{3}{\beta}\) and \(\beta - \frac{3}{\alpha}\)

1. Determine the value of \(A\) [3]
2. Determine the value of \(B\) [3]

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

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

The quadratic equation \(x^2 + 2kx + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac{\alpha + \beta}{\alpha}\) and \(\frac{\alpha + \beta}{\beta}\). [7]

The cubic equation \(z^3 + 4z^2 - 3z + 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma\), \(\alpha\beta + \beta\gamma + \gamma\alpha\) and \(\alpha\beta\gamma\). [3]
2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 22\). [3]

The roots of the equation $$z^3 - 5z^2 + kz - 4 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha\beta\gamma\). [2 marks]
   2. Hence find the value of \(\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2\). [2 marks]
2. The value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\) is \(-4\).
   1. Explain why \(\alpha\), \(\beta\) and \(\gamma\) cannot all be real. [1 mark]
   2. By considering \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\), find the possible values of \(k\). [4 marks]

The cubic equation $$z^3 - 2z^2 + k = 0 \quad (k \neq 0)$$ has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha\beta + \beta\gamma + \gamma\alpha\). [2 marks]
   2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 4\). [2 marks]
   3. Explain why \(\alpha^3 - 2\alpha^2 + k = 0\). [1 mark]
   4. Show that \(\alpha^3 + \beta^3 + \gamma^3 = 8 - 3k\). [2 marks]
2. Given that \(\alpha^4 + \beta^4 + \gamma^4 = 0\):
   1. show that \(k = 2\); [4 marks]
   2. find the value of \(\alpha^5 + \beta^5 + \gamma^5\). [3 marks]

The roots of the equation \(z^3 - 1 = 0\) are denoted by \(1, \omega\) and \(\omega^2\).

1. Sketch an Argand diagram to show these roots. [1]
2. Show that \(1 + \omega + \omega^2 = 0\). [2]
3. Hence evaluate
   1. \((2 + \omega)(2 + \omega^2)\), [2]
   2. \(\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}\). [2]
4. Hence find a cubic equation, with integer coefficients, which has roots \(2, \frac{1}{2 + \omega}\) and \(\frac{1}{2 + \omega^2}\). [4]

\(\alpha\), \(\beta\) and \(\gamma\) are the real roots of the cubic equation $$x^3 + mx^2 + nx + 2 = 0$$ By considering \((\alpha - \beta)^2 + (\gamma - \alpha)^2 + (\beta - \gamma)^2\), prove that $$m^2 \geq 3n$$ [4 marks]

The graph of \(y = x^3 - 3x\) is shown below. \includegraphics{figure_14} The two stationary points have \(x\)-coordinates of \(-1\) and \(1\) The cubic equation $$x^3 - 3x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha\), \(\beta\) and \(\gamma\). The roots \(\alpha\) and \(\beta\) are not real.

1. Explain why \(\alpha + \beta = -\gamma\) [1 mark]
2. Find the set of possible values for the real constant \(p\). [2 marks]
3. \(f(x) = 0\) is a cubic equation with roots \(\alpha + 1\), \(\beta + 1\) and \(\gamma + 1\)
   1. Show that the constant term of \(f(x)\) is \(p + 2\) [3 marks]
   2. Write down the \(x\)-coordinates of the stationary points of \(y = f(x)\) [1 mark]

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 equation \(z^3 + kz^2 + 9 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = k^2$$ [3 marks]
   2. Show that $$\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -18k$$ [4 marks]
2. The equation \(9z^3 - 40z^2 + rz + s = 0\) has roots \(\alpha\beta + \gamma\), \(\beta\gamma + \alpha\) and \(\gamma\alpha + \beta\).
   1. Show that $$k = -\frac{40}{9}$$ [1 mark]
   2. Without calculating the values of \(\alpha\), \(\beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer. [6 marks]

The Argand diagram shows the solutions to the equation \(z^5 = 1\) \includegraphics{figure_3}

1. Solve the equation $$z^5 = 1$$ giving your answers in the form \(z = \cos\theta + i\sin\theta\), where \(0 \leq \theta < 2\pi\) [2 marks]
2. Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon. [2 marks]
3. Show that if \(c = \cos\theta\), where \(z = \cos\theta + i\sin\theta\) is a solution to the equation \(z^5 = 1\), then \(c\) satisfies the equation $$16c^5 - 20c^3 + 5c - 1 = 0$$ [5 marks]
4. The Argand diagram on page 22 is repeated below. \includegraphics{figure_4} Explain, with reference to the Argand diagram, why the expression $$16c^5 - 20c^3 + 5c - 1$$ has a repeated quadratic factor. [3 marks]
5. \(O\) is the centre of a regular pentagon \(ABCDE\) such that \(OA = OB = OC = OD = OE = 1\) unit. The distance from \(O\) to \(AB\) is \(h\) By solving the equation \(16c^5 - 20c^3 + 5c - 1 = 0\), show that $$h = \frac{\sqrt{5} + 1}{4}$$ [5 marks]

The roots of the equation \(20x^3 - 16x^2 - 4x + 7 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\) Find the value of \(\alpha\beta + \beta\gamma + \gamma\alpha\) Circle your answer. [1 mark] \(-\frac{4}{5}\) \(-\frac{1}{5}\) \(\frac{1}{5}\) \(\frac{4}{5}\)

The quadratic equation \(z^2 - 5z + 8 = 0\) has roots \(\alpha\) and \(\beta\)

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\) [2 marks]
2. Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha^4 + \beta^4 = -47\) [4 marks]
3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha^3 + \beta\) and \(\beta^3 + \alpha\) [5 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]

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

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]