4.05a Roots and coefficients: symmetric functions

2 The cubic equation \(6 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } - 3 \mathrm { x } - 5 = 0\), where \(p\) is a constant, has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\).
2. It is given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 ( \alpha + \beta + \gamma )\).
   1. Find the value of \(p\).
   2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

1 The cubic equation \(7 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { - 1 } , \beta ^ { - 1 } , \gamma ^ { - 1 }\).
2. Find the value of \(\alpha ^ { - 2 } + \beta ^ { - 2 } + \gamma ^ { - 2 }\).
3. Find the value of \(\alpha ^ { - 3 } + \beta ^ { - 3 } + \gamma ^ { - 3 }\).

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

1. Find a quartic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 } , \delta ^ { 3 }\) and state the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 }\). [4]
2. Find the value of \(\frac { 1 } { \alpha ^ { 3 } } + \frac { 1 } { \beta ^ { 3 } } + \frac { 1 } { \gamma ^ { 3 } } + \frac { 1 } { \delta ^ { 3 } }\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

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 \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).

4 The cubic equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 6 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\frac { 1 } { \alpha ^ { 3 } } , \frac { 1 } { \beta ^ { 3 } } , \frac { 1 } { \gamma ^ { 3 } }\).
2. Find the value of \(\frac { 1 } { \alpha ^ { 6 } } + \frac { 1 } { \beta ^ { 6 } } + \frac { 1 } { \gamma ^ { 6 } }\).
3. Find also the value of \(\frac { 1 } { \alpha ^ { 9 } } + \frac { 1 } { \beta ^ { 9 } } + \frac { 1 } { \gamma ^ { 9 } }\).

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. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

3 The equation \(x ^ { 4 } - x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).

1. Find a quartic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 } , \delta ^ { 2 }\) and state the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\).
2. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

1 The cubic equation \(2 x ^ { 3 } + x ^ { 2 } - p x - 5 = 0\), where \(p\) is a positive constant, has roots \(\alpha , \beta , \gamma\).

1. State, in terms of \(p\), the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
2. Find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
3. Deduce a cubic equation whose roots are \(\alpha \beta , \beta \gamma , \alpha \gamma\).
4. Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = \frac { 1 } { 3 }\), find the value of \(p\).

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

1. Find a cubic equation whose roots are \(\alpha ^ { 2 } + 1 , \beta ^ { 2 } + 1 , \gamma ^ { 2 } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-04_2714_34_143_2012}
2. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 2 } + \left( \beta ^ { 2 } + 1 \right) ^ { 2 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 2 }\).
3. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 3 } + \left( \beta ^ { 2 } + 1 \right) ^ { 3 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 3 }\).

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.

1 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\), where \(b , c\) and \(d\) are constants, has roots \(\alpha , \beta , \gamma\). It is given that \(\alpha \beta \gamma = - 1\).

1. State the value of \(d\).
2. Find a cubic equation, with coefficients in terms of \(b\) and \(c\), whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\).
3. Given also that \(\gamma + 1 = - \alpha - 1\), deduce that \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).

1 It is given that $$\alpha + \beta + \gamma = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 5 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 6 .$$ The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\) has roots \(\alpha , \beta , \gamma\).
Find the values of \(b , c\) and \(d\).

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.

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 \(d\).

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

1. Find a quartic equation whose roots are \(\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } } , \frac { 1 } { \gamma ^ { 2 } } , \frac { 1 } { \delta ^ { 2 } }\) and state the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
2. Find the value of \(\beta ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
3. Find the value of \(\frac { 1 } { \alpha ^ { 4 } } + \frac { 1 } { \beta ^ { 4 } } + \frac { 1 } { \gamma ^ { 4 } } + \frac { 1 } { \delta ^ { 4 } }\).

3 The quartic equation \(\mathrm { x } ^ { 4 } + \mathrm { bx } ^ { 3 } + \mathrm { cx } ^ { 2 } + \mathrm { dx } - 2 = 0\) has roots \(\alpha , \beta , \gamma , \delta\). It is given that $$\alpha + \beta + \gamma + \delta = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 5 , \quad \alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { - 1 } + \delta ^ { - 1 } = 6$$

1. Find the values of \(b , c\) and \(d\).
2. Given also that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = - 27\), find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

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 \(S _ { - 2 }\).

3 It is given that $$\begin{aligned} & \alpha + \beta + \gamma + \delta = 2 \\ & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 3 \\ & \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = 4 \end{aligned}$$

1. Find the value of \(\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta\).
2. Find the value of \(\alpha ^ { 2 } \beta + \alpha ^ { 2 } \gamma + \alpha ^ { 2 } \delta + \beta ^ { 2 } \alpha + \beta ^ { 2 } \gamma + \beta ^ { 2 } \delta + \gamma ^ { 2 } \alpha + \gamma ^ { 2 } \beta + \gamma ^ { 2 } \delta + \delta ^ { 2 } \alpha + \delta ^ { 2 } \beta + \delta ^ { 2 } \gamma\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-06_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-07_2723_33_99_22}
3. It is given that \(\alpha , \beta , \gamma , \delta\) are the roots of the equation $$6 x ^ { 4 } - 12 x ^ { 3 } + 3 x ^ { 2 } + 2 x + 6 = 0 .$$
   1. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
   2. Find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }\).

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.

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}

5. The quadratic equation $$4 x ^ { 2 } + 3 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\).
2. Find the value of \(\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)\).
3. Find a quadratic equation which has roots $$( 4 \alpha - \beta ) \text { and } ( 4 \beta - \alpha )$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.

3. The quadratic equation $$x ^ { 2 } - 2 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,

1. 1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
   2. show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 2\)
   3. find the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. 1. show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }\)
   2. find a quadratic equation which has roots $$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.

1. The quadratic equation

$$3 x ^ { 2 } + 2 x + 5 = 0$$ has roots \(\alpha\) and \(\beta\). Without solving the equation,

1. find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
2. show that \(\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }\)
3. find a quadratic equation which has roots $$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.

1. The equation \(2 x ^ { 2 } + 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\)

Without solving the equation

1. determine the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. form a quadratic equation, with integer coefficients, which has roots $$\frac { \alpha ^ { 2 } } { \beta } \text { and } \frac { \beta ^ { 2 } } { \alpha }$$ \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-09_2255_50_314_34}
   

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