4.05a Roots and coefficients: symmetric functions

8 The equation \(4 \mathrm { x } ^ { 4 } - 4 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , - \alpha , \beta\) and \(\frac { 1 } { \beta }\).

1. Determine the exact roots of the equation.
2. Determine the values of \(p\) and \(q\).

3 In the first week of an outbreak of influenza, 9 patients were diagnosed with the virus at a medical practice in Pencaster. Records were kept of \(y\), the total number of patients diagnosed with influenza in week \(n\). The data are shown in Fig. 3. \begin{table}[h] \end{table}

1. Complete the difference table in the Printed Answer Booklet.
2. Explain why a cubic model is appropriate for the data.
3. Use Newton's method to find the interpolating polynomial of degree 3 for these data. In both week 6 and week 7 there were 145 patients in total diagnosed with influenza at the medical practice.
4. Determine whether the model is a good fit for these data.
5. Determine the maximum number of weeks for which the model could possibly be valid.

7 Sam decided to go on a high-protein diet. Sam's mass in \(\mathrm { kg } , M\), after \(t\) days of following the diet is recorded in Fig. 7.1. \begin{table}[h] \end{table} A difference table for the data is shown in Fig. 7.2. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet. Sam's doctor uses these data to construct a cubic interpolating polynomial to model Sam's mass at time \(t\) days after starting the diet.
2. Find the model in the form \(\mathrm { M } = \mathrm { at } ^ { 3 } + \mathrm { bt } ^ { 2 } + \mathrm { ct } + \mathrm { d }\), where \(a , b , c\) and \(d\) are constants to be determined. Subsequently it is found that when \(\mathrm { t } = 40 , \mathrm { M } = 78.7\) and when \(\mathrm { t } = 50 , \mathrm { M } = 80.05\).
3. Determine whether the model is a good fit for these data.
4. By completing the extended copy of Fig. 7.2 in the Printed Answer Booklet, explain why a quartic model may be more appropriate for the data.
5. Refine the doctor's model to include a quartic term.
6. Explain whether the new model for Sam's mass is likely to be appropriate over a longer period of time.

2 A car tyre has a slow puncture. Initially the tyre is inflated to a pressure of 34.5 psi . The pressure is checked after 3 days and then again after 5 days. The time \(t\) in days and the pressure, \(P\) psi, are shown in the table below. You are given that the pressure in a car tyre is measured in pounds per square inch (psi). The owner of the car believes the relationship between \(P\) and \(t\) may be modelled by a polynomial.

1. Explain why it is not possible to use Newton's forward difference interpolation method for these data.
2. Use Lagrange's form of the interpolating polynomial to find an interpolating polynomial of degree 2 for these data. The car owner uses the polynomial found in part (b) to model the relationship between \(P\) and \(t\).
   Subsequently it is found that when \(t = 6 , P = 26.0\) and when \(t = 10 , P = 24.4\).
3. Determine whether the owner's model is a good fit for these data.
4. Explain why the model would not be suitable in the long term.

4 Between 1946 and 2012 the mean monthly maximum temperature of the water surface of a lake in northern England has been recorded by environmental scientists. Some of the data are shown in Table 4.1. \begin{table}[h] \end{table} Table 4.2 shows a difference table for the data. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet.
2. Explain why a quadratic model may be appropriate for these data.
3. Use Newton's forward difference interpolation formula to construct an interpolating polynomial of degree 2 for these data. This polynomial is used to model the relationship between \(T\) and \(t\). Between 1946 and 2012 the mean monthly maximum temperature of the water surface of the lake was recorded as \(8.9 ^ { \circ } \mathrm { C }\) for October and \(7.5 ^ { \circ } \mathrm { C }\) for November.
4. Determine whether the model is a good fit for the temperatures recorded in October and November. A scientist recorded the mean monthly maximum temperature of the water surface of the lake in 2022. Some of the data are shown in Table 4.3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.3}

   Month	May	June	July	August	September
   \(t =\) Time in months	0	1	2	3	4
   \(T =\) Mean temperature in \({ } ^ { \circ } \mathrm { C }\)	10.3	14.7	16.9	16.9	14.8

   \end{table}
5. Adapt the polynomial found in part (c) so that it can be used to model the relationship between \(T\) and \(t\) for the data in Table 4.3.

5. Given that \(x = - \frac { 1 } { 2 }\) and \(x = - 3\) are two roots of the equation $$2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 23 x + 15 = 0$$ find the remaining roots.

10. The quadratic equation \(p x ^ { 2 } + q x + r = 0\) has roots \(\alpha\) and \(\beta\), where \(p , q , r\) are non-zero constants.

1. A cubic equation is formed with roots \(\alpha , \beta , \alpha + \beta\). Find the cubic equation with coefficients expressed in terms of \(p , q , r\).
2. Another quadratic equation \(p x ^ { 2 } - q x - r = 0\) has roots \(2 \alpha\) and \(\gamma\). Show that \(\beta = - 2 \gamma\).

6. The roots of the cubic equation $$2 x ^ { 3 } + p x ^ { 2 } - 126 x + q = 0$$ form a geometric progression with common ratio - 3 .
Find the possible values of \(p\) and \(q\), giving your answers in surd form.

8. The roots of the cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 2 x + 8 = 0\) are denoted by \(\alpha , \beta , \gamma\). Determine the cubic equation whose roots are \(\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }\).
Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), where \(a , b , c , d\) are constants to be determined.

3. The quadratic equation \(x ^ { 2 } + p x + q = 0\) has a repeated root \(\alpha\). A new quadratic equation has a repeated root \(\frac { 1 } { \alpha }\) and is of the form \(x ^ { 2 } + m x + m = 0\).
Find the values of \(p\) and \(q\) in the original equation.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).

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

1. solve completely the equation \(\mathrm { f } ( z ) = 0\)
2. Hence find the value of \(p\).

7. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has complex roots \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) and \(\mathrm { z } _ { 4 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 } , z _ { 3 }\) and \(z _ { 4 }\) form the vertices of a square, with one vertex in each quadrant.
Given that \(z _ { 1 } = 2 + 3 i\), determine the values of \(a , b , c\) and \(d\).

1. The cubic equation

$$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha , \beta\), and \(\gamma\).
Without solving the cubic equation,

1. determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
2. find a cubic equation that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\), giving your answer in the form \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers to be determined.

1. The roots of the quartic equation

$$3 x ^ { 4 } + 5 x ^ { 3 } - 7 x + 6 = 0$$ are \(\alpha , \beta , \gamma\) and \(\delta\) Making your method clear and without solving the equation, determine the exact value of

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\)
2. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma } + \frac { 2 } { \delta }\)
3. \(( 3 - \alpha ) ( 3 - \beta ) ( 3 - \gamma ) ( 3 - \delta )\)

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. The quartic equation $$z ^ { 4 } + 5 z ^ { 2 } - 30 = 0$$ has roots \(p , q , r\) and \(s\).
   Without solving the equation, determine the quartic equation whose roots are $$( 3 p - 1 ) , ( 3 q - 1 ) , ( 3 r - 1 ) \text { and } ( 3 s - 1 )$$ Give your answer in the form \(w ^ { 4 } + a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be found.
2. The roots of the cubic equation $$4 x ^ { 3 } + n x + 81 = 0 \quad \text { where } n \text { is a real constant }$$ are \(\alpha , 2 \alpha\) and \(\alpha - \beta\) Determine

1. the values of the roots of the equation,
2. the value of \(n\).

1. The cubic equation

$$2 x ^ { 3 } - 3 x ^ { 2 } + 5 x + 7 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, determine the exact value of

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)
2. \(\frac { 3 } { \alpha } + \frac { 3 } { \beta } + \frac { 3 } { \gamma }\)
3. \(( 5 - \alpha ) ( 5 - \beta ) ( 5 - \gamma )\)

1. $$f ( z ) = z ^ { 3 } + p z ^ { 2 } + q z - 15$$ where \(p\) and \(q\) are real constants.
Given that the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has roots $$\alpha , \frac { 5 } { \alpha } \text { and } \left( \alpha + \frac { 5 } { \alpha } - 1 \right)$$

1. solve completely the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
2. Hence find the value of \(p\).

3 A tree at the bottom of a garden needs to be reduced in height. The tree is known to increase in height by 15 centimetres each year. On the first day of every year, the height is measured and the tree is immediately trimmed by \(3 \%\) of this height. When the tree is measured, before trimming on the first day of year 1 , the height is 6 metres.
Let \(H _ { n }\) be the height of the tree immediately before trimming on the first day of year \(n\).

1. Explain, in the context of the problem, why the height of the tree may be modelled by the recurrence relation $$H _ { n + 1 } = 0.97 H _ { n } + 0.15 , \quad H _ { 1 } = 6 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(H _ { n } = 0.97 ^ { n - 1 } + 5 , \quad n \geqslant 1\)
3. Explain what will happen to the height of the tree immediately before trimming in the long term.
4. By what fixed percentage should the tree be trimmed each year if the height of the tree immediately before trimming is to be 4 metres in the long term?

1. On Jim's 11 th birthday his parents invest \(\pounds 1000\) for him in a savings account.

The account earns 2\% interest each year.
On each subsequent birthday, Jim's parents add another \(\pounds 500\) to this savings account.
Let \(U _ { n }\) be the amount of money that Jim has in his savings account \(n\) years after his 11th birthday, once the interest for the previous year has been paid and the \(\pounds 500\) has been added.

1. Explain, in the context of the problem, why the amount of money that Jim has in his savings account can be modelled by the recurrence relation of the form $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
2. State an assumption that must be made for this model to be valid.
3. Solve the recurrence relation $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ Jim hopes to be able to buy a car on his 18th birthday.
4. Use the answer to part (c) to find out whether Jim will have enough money in his savings account to buy a car that costs \(\pounds 4500\)

1. The roots of the equation

$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$ are \(\alpha , \beta\) and \(\gamma\) Without solving the equation,

1. write down the value of each of $$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
2. Use the answers to part (a) to determine the value of
   1. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }\)
   2. \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)
   3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)

1. The roots of the equation

$$x ^ { 3 } - 2 x ^ { 2 } + 4 x - 5 = 0$$ are \(p , q\) and \(r\).
Without solving the equation, find the value of

1. \(\frac { 2 } { p } + \frac { 2 } { q } + \frac { 2 } { r }\)
2. \(( p - 4 ) ( q - 4 ) ( r - 4 )\)
3. \(p ^ { 3 } + q ^ { 3 } + r ^ { 3 }\)

1. The cubic equation

$$4 x ^ { 3 } + p x ^ { 2 } - 14 x + q = 0$$ where \(p\) and \(q\) are real positive constants, has roots \(\alpha , \beta\) and \(\gamma\) Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 16\)

1. show that \(p = 12\) Given that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } = \frac { 14 } { 3 }\)
2. determine the value of \(q\) Without solving the cubic equation,
3. determine the value of \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)

1. The roots of the equation

$$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$ are \(\alpha , \beta\) and \(\gamma\) Without solving the equation, find the value of

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

4 In this question you must show detailed reasoning. The roots of the cubic equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { 2 } ( \alpha - 1 ) , \frac { 1 } { 2 } ( \beta - 1 )\) and \(\frac { 1 } { 2 } ( \gamma - 1 )\).
2. Hence or otherwise solve the equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).