4.05a Roots and coefficients: symmetric functions

In this question, the argument of a complex number is defined as being in the range \([0, 2\pi)\). You are given that \(\omega_k\), where \(k = 0, 1, 2, ..., n-1\), are the \(n\) \(n\)th roots of unity for some integer \(n\), \(n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega_0 = 1\)).

1. With the help of a diagram explain why \(\omega_k = (\omega_1)^k\) for \(k = 2, ..., n-1\). [3]
2. Using the identity given in part (a), show that \(\sum_{k=0}^{n-1}\omega_k = 0\). [2]
3. Show that if \(z\) is a complex number then \(z + z^* = 2\text{Re}(z)\). [1]
4. Using the results from parts (b) and (c) show that \(\sum_{k=0}^{n-1}\text{Re}(\omega_k) = 0\). [1]
5. With the help of a diagram explain why \(\text{Re}(\omega_k) = \text{Re}(\omega_{n-k})\) for \(k = 1, 2, ..., n-1\). [1]

You should now consider the case when \(n = 5\).

1. 1. Use parts (d) and (e) to deduce that \(\cos\frac{4\pi}{5} = a + b\cos\frac{2\pi}{5}\), for some rational constants \(a\) and \(b\). [2]
   2. Hence determine the exact value of \(\cos\frac{2\pi}{5}\). [2]

The cubic equation \(x^3 - 4x^2 + px + q = 0\) has roots \(\alpha\), \(\frac{2}{\alpha}\) and \(\alpha + \frac{2}{\alpha}\). Find

• the values of the roots of the equation,
• the value of \(p\).

[7]

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 roots of the equation $$x^3 - 4x^2 + 14x - 20 = 0$$ are denoted by \(\alpha\), \(\beta\), \(\gamma\).

1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = -12.$$ Explain why this result shows that exactly one of the roots of the above cubic equation is real. [3]
2. Given that one of the roots is \(1 + 3i\), find the other two roots. Explain your method for each root. [4]

In this question you must show detailed reasoning. You are given that \(\alpha\), \(\beta\) and \(\gamma\) are the roots of the equation \(5x^3 - 2x^2 + 3x + 1 = 0\).

1. Find the value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\). [5]
2. Find a cubic equation whose roots are \(\alpha^2\), \(\beta^2\) and \(\gamma^2\) giving your answer in the form \(ax^3 + bx^2 + cx + d = 0\) where \(a\), \(b\), \(c\) and \(d\) are integers. [4]

In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
2. Given that \(\alpha^3\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^3\), \(\beta^3\) and \(\gamma^3\). [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]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The roots of the equation $$x^3 - 8x^2 + 28x - 32 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the value of $$(\alpha + 2)(\beta + 2)(\gamma + 2).$$ [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]

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]

$$f(z) = z^3 - 8z^2 + pz - 24$$ where \(p\) is a real constant. Given that the equation \(f(z) = 0\) has distinct roots $$\alpha, \beta \text{ and } \left(\alpha + \frac{12}{\alpha} - \beta\right)$$

1. solve completely the equation \(f(z) = 0\) [6]
2. Hence find the value of \(p\). [2]

The quartic equation $$2x^4 + Ax^3 - Ax^2 - 5x + 6 = 0$$ where \(A\) is a real constant, has roots \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\)

1. Determine the value of $$\frac{3}{\alpha} + \frac{3}{\beta} + \frac{3}{\gamma} + \frac{3}{\delta}$$ [3]

Given that \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = -\frac{3}{4}\)

1. determine the possible values of \(A\) [5]

In this question you must show detailed reasoning. The distinct numbers \(\omega_1\) and \(\omega_2\) both satisfy the quadratic equation \(4x^2 + 4x + 17 = 0\).

1. Write down the value of \(\omega_1 \omega_2\). [1]
2. \(A\), \(B\) and \(C\) are the points on an Argand diagram which represent \(\omega_1\), \(\omega_2\) and \(\omega_1 \omega_2\). Find the area of triangle \(ABC\). [6]

In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 \equiv (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
2. Given that \(\alpha^2\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^2\), \(\beta^3\) and \(\gamma^3\). [4]

You are given that the cubic equation \(2x^3 - 3x^2 + x + 4 = 0\) has three roots, \(\alpha\), \(\beta\) and \(\gamma\). By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]

The roots of the equation \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are positive integers, are \(\alpha\) and \(\beta\).

1. Find a quadratic equation with integer coefficients whose roots are \(\alpha + \beta\) and \(\alpha\beta\). [4]
2. Show that it is not possible for the original equation and the equation found in part (i) both to have repeated roots. [2]
3. Show that the discriminant of the equation found in part (i) is always positive. [3]

**In this question you must show detailed reasoning.** The equation \(x^2 + 2x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x^2 + px + q = 0\) has roots \(\alpha^2\) and \(\beta^2\). Find the values of \(p\) and \(q\). [3]

The cubic equation \(x^3 + x^2 + 7x - 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Show that \(\alpha^2 + \beta^2 + \gamma^2 = -13\). [3]
2. State what can be deduced about the nature of these roots. [2]

The roots of the equation \(x^4 - 2x^3 + 2x^2 + x - 3 = 0\) are \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Determine the values of

1. \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\), [2]
2. \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}\), [2]
3. \(\alpha^3 + \beta^3 + \gamma^3 + \delta^3\). [4]

The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).

1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]