4.05a Roots and coefficients: symmetric functions

1. The quadratic equation

$$4 x ^ { 2 } + 3 x + k = 0$$ where \(k\) is an integer, has roots \(\alpha\) and \(\beta\)

1. Write down, in terms of \(k\) where appropriate, the value of \(\alpha + \beta\) and the value of \(\alpha \beta\)
2. Determine, in simplest form in terms of \(k\), the value of \(\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }\)
3. Determine a quadratic equation which has roots $$\frac { \alpha } { \beta ^ { 2 } } \text { and } \frac { \beta } { \alpha ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integer values in terms of \(k\)

1. The quadratic equation

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

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

6. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(3 x ^ { 2 } + 5 x - 1 = 0\)

1. Find the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. Find a quadratic equation which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers.

3. It is given that \(\alpha\) and \(\beta\) are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$

1. Find the exact value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
2. Find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.

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

1. find the exact value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. Find a quadratic equation which has roots ( \(\alpha ^ { 2 } + \beta\) ) and ( \(\beta ^ { 2 } + \alpha\) ), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
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1. The quadratic equation

$$3 x ^ { 2 } - 5 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation, find the exact value of $$\frac { \alpha } { \beta } + \frac { \beta } { \alpha }$$ Count coution \(\_\_\_\_\) T

7. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(5 x ^ { 2 } - 4 x + 3 = 0\) Without solving the quadratic equation,

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

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

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

1. The quadratic equation

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

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

5. $$f ( x ) = x ^ { 2 } - 6 x + 3$$ The equation \(\mathrm { f } ( x ) = 0\) has roots \(\alpha\) and \(\beta\) Without solving the equation,

1. determine the value of $$\left( \alpha ^ { 2 } + 1 \right) \left( \beta ^ { 2 } + 1 \right)$$
2. find a quadratic equation which has roots $$\frac { \alpha } { \left( \alpha ^ { 2 } + 1 \right) } \text { and } \frac { \beta } { \left( \beta ^ { 2 } + 1 \right) }$$ 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 $$2 x ^ { 2 } - 5 x + 7 = 0$$ has roots \(\alpha\) and \(\beta\) Without solving the equation,

1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
2. determine, giving each answer as a simplified fraction, the value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
3. find a quadratic equation that has roots $$\frac { 1 } { \alpha ^ { 2 } + \beta } \text { and } \frac { 1 } { \beta ^ { 2 } + \alpha }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.

1. The quadratic equation

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

1. find the exact value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. Find a quadratic equation which has roots ( \(\alpha ^ { 2 } + \beta\) ) and ( \(\beta ^ { 2 } + \alpha\) ), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.

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1. The quadratic equation

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

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }\)
3. Find a quadratic equation with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$

6. Given that 2 and \(5 + 2 \mathrm { i }\) are roots of the equation $$x ^ { 3 } - 12 x ^ { 2 } + c x + d = 0 , \quad c , d \in \mathbb { R }$$

1. write down the other complex root of the equation.
2. Find the value of \(c\) and the value of \(d\).
3. Show the three roots of this equation on a single Argand diagram.

3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\).
2. Find the sum of these four 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 \(\mathrm { f } ( x ) = 0\).

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

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

10 The roots of the equation $$x ^ { 3 } - 9 x ^ { 2 } + 27 x - 29 = 0$$ are denoted by \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.

1. Write down the value of \(\alpha + \beta + \gamma\).
2. It is given that \(\beta = p + \mathrm { i } q\), where \(q > 0\). Find the value of \(p\), in terms of \(\alpha\).
3. Write down the value of \(\alpha \beta \gamma\).
4. Find the value of \(q\), in terms of \(\alpha\) only.

7 The quadratic equation \(x ^ { 2 } + 5 x + 10 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 5\).
3. Hence find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).

9

1. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )\).
2. The quadratic equation \(x ^ { 2 } - 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\).
3. Show that \(\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
4. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
5. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
6. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }\), find the value of \(N\).

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

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
2. Find the value of \(p\).
3. Find the value of \(q\).

6 The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. (a) Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha \beta + \beta \gamma + \gamma \alpha\).
   (b) Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. (a) Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
   (b) Use your answer to part (ii) (a) to find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).

6 The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are real, has roots ( \(3 + \mathrm { i }\) ) and 2 .

1. Write down the other root of the equation.
2. Find the values of \(a , b\) and \(c\).

8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
Find the value of \(( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )\) in terms of \(k\).