Roots of polynomials

Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]

2. Given that \(- 2 + 3 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are real constants,

1. write down the other root of the equation.
2. Find the value of \(p\) and the value of \(q\).

9 You are given that the cubic equation \(2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0\), where \(p\) and \(q\) are real numbers, has a complex root \(\alpha = 1 + \mathrm { i } \sqrt { 2 }\).

1. Write down a second complex root, \(\beta\).
2. Determine the third root, \(\gamma\).
3. Find the value of \(p\) and the value of \(q\).
4. Show that if \(n\) is an integer then \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }\) where \(\tan \theta = \sqrt { 2 }\).

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 \(3 - 2 \beta\).

2. The quadratic equation $$5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots $$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{4da2bb2c-a51b-493c-a9f2-f4ff008a3aac-07_70_51_2663_1896}

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 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]

**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]

3 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { cx } + 1 = 0\), where \(c\) is a constant, has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\).
2. Show that \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } = 3 - 2 c ^ { 3 }\).
3. Find the real value of \(c\) for which the matrix \(\left( \begin{array} { c c c } 1 & \alpha ^ { 3 } & \beta ^ { 3 } \\ \alpha ^ { 3 } & 1 & \gamma ^ { 3 } \\ \beta ^ { 3 } & \gamma ^ { 3 } & 1 \end{array} \right)\) is singular.

4 The roots of the cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 8 x + 11 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).

6 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 3 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(2 \alpha , 2 \beta\) and \(2 \gamma\), expressing your answer in a form with integer coefficients.

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]

3 The equation \(2 x ^ { 3 } - 2 x ^ { 2 } + 8 x - 15 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Determine the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).

3 The roots of the equation \(x ^ { 2 } - p x - 6 = 0\) are \(\alpha\) and \(\beta\) Find \(\alpha ^ { 2 } + \beta ^ { 2 }\) in terms of \(p\) Circle your answer. \(p ^ { 2 } - 6\) \(p ^ { 2 } + 6\) \(p ^ { 2 } - 12\) \(p ^ { 2 } + 12\)

In the equation $$x^3 + ax^2 + bx + c = 0,$$ the coefficients \(a\), \(b\) and \(c\) are real. It is given that all the roots are real and greater than \(1\).

1. Prove that \(a < -3\). [1]
2. By considering the sum of the squares of the roots, prove that \(a^2 > 2b + 3\). [2]
3. By considering the sum of the cubes of the roots, prove that \(a^3 < -9b - 3c - 3\). [4]

1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).

2 The cubic equation \(x ^ { 3 } - 6 x ^ { 2 } + k x + 10 = 0\) has roots \(p - q , p\) and \(p + q\), where \(q\) is positive.

1. By considering the sum of the roots, find \(p\).
2. Hence, by considering the product of the roots, find \(q\).
3. Find the value of \(k\).

1 The cubic equation \(x ^ { 3 } + b x ^ { 2 } + d = 0\) has roots \(\alpha , \beta , \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).

1. Show that \(4 b ^ { 3 } + 27 d = 0\).
2. Given that \(2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b\), find the values of \(b\) and \(d\).

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

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]

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 \(S _ { 8 }\).

2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\) Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
[0pt] [1 mark] \(p - p - q - q\)

2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\) Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
[0pt] [1 mark] \(p - p - q - q\)

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]

2 Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.

4. $$f ( x ) = x ^ { 3 } + x ^ { 2 } + 44 x + 150$$ Given that \(\mathrm { f } ( x ) = ( x + 3 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Find the three roots of \(\mathrm { f } ( x ) = 0\).
3. Find the sum of the three roots of \(\mathrm { f } ( x ) = 0\).

10 The polynomial \(x ^ { 3 } + 5 x ^ { 2 } + 31 x + 75\) is denoted by \(\mathrm { p } ( x )\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).
2. Show that \(z = - 1 + 2 \sqrt { 6 } \mathrm { i }\) is a root of \(\mathrm { p } ( z ) = 0\).
3. Hence find the complex numbers \(z\) which are roots of \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 Find a cubic equation with roots \(\alpha , \beta\) and \(\gamma\), given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.

5 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha \beta \gamma & = - 7 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 13 \end{aligned}$$

1. Write down the values of \(p\) and \(r\).
2. Find the value of \(q\).

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]

3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).

3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).

\(\alpha\), \(\beta\) 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]

4 The roots of the quadratic equation \(x ^ { 2 } + x - 8 = 0\) are \(p\) and \(q\). Find the value of \(p + q + \frac { 1 } { p } + \frac { 1 } { q }\).

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]

4 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).

3 The equation $$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$ has roots \(\alpha , \beta , \gamma\). Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\). Hence show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{array} \right)\) is singular.