Sum of powers of roots

2 The cubic equation \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) has roots \(\alpha , \beta , \gamma\). Let \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } } + \gamma ^ { \mathrm { n } }\).

1. State the value of \(S _ { 1 }\) and find the value of \(S _ { 2 }\).
2. 1. Express \(\mathrm { S } _ { \mathrm { n } + 3 }\) in terms of \(\mathrm { S } _ { \mathrm { n } + 2 }\) and \(\mathrm { S } _ { \mathrm { n } }\).
   2. Hence, or otherwise, find the value of \(S _ { 4 }\).
3. Use the substitution \(\mathrm { y } = \mathrm { S } _ { 1 } - \mathrm { x }\), where \(S _ { 1 }\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
4. Find the value of \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).

2 The cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 10 x - 2 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_58_1550_397_347} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_67_1566_481_328} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_65_1566_573_328} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_65_1570_662_324} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_65_1570_751_324} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_67_1570_840_324} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_68_1570_931_324} \includegraphics[max width=\textwidth, alt={}]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_63_1570_1023_324} .......................................................................................................................................... . .......................................................................................................................................... ......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_72_1570_1379_324} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_80_1570_1556_324} ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_72_1572_1740_322} ....................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_74_1572_1921_322} \includegraphics[max width=\textwidth, alt={}, center]{e34e64b7-d3bb-4614-bf37-5ef90ad56157-04_70_1570_2012_324}
2. Show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{array} \right)\) is singular.

2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 1\).
3. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 3 } + ( \beta + r ) ^ { 3 } + ( \gamma + r ) ^ { 3 } \right) = n + \frac { 1 } { 4 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are constants to be determined.

4 The cubic equation \(27 x ^ { 3 } + 18 x ^ { 2 } + 6 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Show that a cubic equation with roots \(3 \alpha + 1,3 \beta + 1,3 \gamma + 1\) is $$y ^ { 3 } - y ^ { 2 } + y - 2 = 0$$ The sum \(( 3 \alpha + 1 ) ^ { n } + ( 3 \beta + 1 ) ^ { n } + ( 3 \gamma + 1 ) ^ { n }\) is denoted by \(\mathrm { S } _ { \mathrm { n } }\).
2. Find the values of \(S _ { 2 }\) and \(S _ { 3 }\).
3. Find the values of \(S _ { - 1 }\) and \(S _ { - 2 }\).

4 The cubic equation $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Show that the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\) is 19 .
2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }\).
3. Find a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), giving your answer in the form $$p x ^ { 3 } + q x ^ { 2 } + r x + s = 0 ,$$ where \(p , q , r\) and \(s\) are constants to be determined.

1 The roots of the cubic equation \(x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0\) are \(\alpha , \beta , \gamma\). Find the values of

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\),
2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

The roots of the equation $$x ^ { 4 } - 5 x ^ { 2 } + 2 x - 1 = 0$$ are \(\alpha , \beta , \gamma , \delta\). Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\).

1. Show that $$S _ { n + 4 } - 5 S _ { n + 2 } + 2 S _ { n + 1 } - S _ { n } = 0 .$$
2. Find the values of \(S _ { 2 }\) and \(S _ { 4 }\).
3. Find the value of \(S _ { 3 }\) and hence find the value of \(S _ { 6 }\).
4. Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 } \right) + \beta ^ { 2 } \left( \gamma ^ { 4 } + \delta ^ { 4 } + \alpha ^ { 4 } \right) + \gamma ^ { 2 } \left( \delta ^ { 4 } + \alpha ^ { 4 } + \beta ^ { 4 } \right) + \delta ^ { 2 } \left( \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } \right) .$$

The roots of the equation \(x ^ { 4 } - 3 x ^ { 2 } + 5 x - 2 = 0\) are \(\alpha , \beta , \gamma , \delta\), and \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). Show that $$S _ { n + 4 } - 3 S _ { n + 2 } + 5 S _ { n + 1 } - 2 S _ { n } = 0$$ Find the values of

1. \(S _ { 2 }\) and \(S _ { 4 }\),
2. \(S _ { 3 }\) and \(S _ { 5 }\). Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } \right) + \beta ^ { 2 } \left( \gamma ^ { 3 } + \delta ^ { 3 } + \alpha ^ { 3 } \right) + \gamma ^ { 2 } \left( \delta ^ { 3 } + \alpha ^ { 3 } + \beta ^ { 3 } \right) + \delta ^ { 2 } \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) .$$

2 The cubic equation \(x ^ { 3 } - p x - q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , \beta , \gamma\). Show that

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 p\),
2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 q\),
3. \(6 \left( \alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } \right) = 5 \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } \right)\).

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 1\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 }\).

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. \(p ^ { 2 } - 6\) \(p ^ { 2 } + 6\) \(p ^ { 2 } - 12\) \(p ^ { 2 } + 12\)

2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined.

7

1. Express \(x ^ { 3 } + y ^ { 3 }\) in terms of \(( x + y )\) and \(x y\).
2. The equation \(t ^ { 2 } - 3 t + \frac { 8 } { 9 } = 0\) has roots \(\alpha\) and \(\beta\).
   1. Determine the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\).
   2. Hence express 19 as the sum of the cubes of two positive rational numbers.

2 The equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Evaluate \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and use your answer to comment on the nature of these roots.

2 The equation \(x ^ { 3 } - 14 x ^ { 2 } + 16 x + 21 = 0\) has roots \(\alpha , \beta , \gamma\). Determine the values of \(\alpha + \beta + \gamma\), \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

Find the sum of the squares of the roots of the equation $$x^3 + x + 12 = 0,$$ and deduce that only one of the roots is real. [4] The real root of the equation is denoted by \(\alpha\). Prove that \(-3 < \alpha < -2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt{6}\). [5]

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]

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

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 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]

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]

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]