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 \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
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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 \times 3 ^ { n } & 0
3 ^ { n } - 1 & 2 \end{array} \right)$$ (b) Find, in terms of \(n\), the inverse of \(\mathbf { A } ^ { n }\).\\ 2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).\\ (a) Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).\\ (b) 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 Th cb c ę tin $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ h s ro \(\mathrm { s } \alpha , \beta\) ad \(\gamma\).

1. Sth th t th le \(6 \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\) is 9
2. Fid he le \(6 \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }\).
3. Fird cb ceq tin \(N\) ith o s \(\alpha + 1 \beta + 1 \mathrm {~d} \gamma + \underset { \text { g } } { \text { vg } } \quad\) as wer in th fo m $$p x ^ { 3 } + q x ^ { 2 } + r x + s = 0$$ we re \(p , q , r\) ad \(s\) are co tan s to \(\mathbf { b } \quad \mathbf { d }\) termin d

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 $$\alpha ^ { 2 } \left( \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 } \right) + \beta ^ { 2 } \left( \gamma ^ { 4 } + \delta ^ { 4 } + \alpha ^ { 4 } \right) + \gamma ^ { 2 } \left( \delta ^ { 4 } + \alpha ^ { 4 } + \beta ^ { 4 } \right) + \delta ^ { 2 } \left( \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } \right) .$$ \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 $$\alpha ^ { 2 } \left( \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } \right) + \beta ^ { 2 } \left( \gamma ^ { 3 } + \delta ^ { 3 } + \alpha ^ { 3 } \right) + \gamma ^ { 2 } \left( \delta ^ { 3 } + \alpha ^ { 3 } + \beta ^ { 3 } \right) + \delta ^ { 2 } \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) .$$

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

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

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

1. Show that $$( r + 1 ) ! - ( r - 1 ) ! = \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) !$$
2. Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) ! = ( n + 2 ) n ! - 2$$ (4 marks) The cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + k = 0 \quad ( k \neq 0 )$$ has roots \(\alpha , \beta\) and \(\gamma\).
3. 1. Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha \beta + \beta \gamma + \gamma \alpha\).
   2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 4\).
   3. Explain why \(\alpha ^ { 3 } - 2 \alpha ^ { 2 } + k = 0\).
   4. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 8 - 3 k\).
4. Given that \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } = 0\) :
   1. show that \(k = 2\);
   2. find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 }\).

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

13 The quadratic equation \(z ^ { 2 } - 5 z + 8 = 0\) has roots \(\alpha\) and \(\beta\) 13

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\)
   13
2. Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha ^ { 4 } + \beta ^ { 4 } = - 47\)
   [0pt] [4 marks]
   
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