Roots of polynomials

4 The roots of the cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 8 x + 11 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).

6 The cubic equation $$x ^ { 3 } + 5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\)
Find a cubic equation, with integer coefficients, whose roots are \(3 \alpha , 3 \beta\) and \(3 \gamma\)

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.

3 You are given that \(z = 2 + 3 \mathrm { j }\) is a root of the quartic equation \(z ^ { 4 } - 5 z ^ { 3 } + 15 z ^ { 2 } - 5 z - 26 = 0\). Find the other roots.

4 The function f is a quartic function with real coefficients.
The complex number 5i is a root of the equation \(\mathrm { f } ( x ) = 0\)
Which one of the following must be a factor of \(\mathrm { f } ( x )\) ?
Circle your answer.
( \(x ^ { 2 } - 25\) )
\(\left( x ^ { 2 } - 5 \right)\)
\(\left( x ^ { 2 } + 5 \right)\)
\(\left( x ^ { 2 } + 25 \right)\)

9 You are given that the cubic equation \(2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0\), where \(p\) and \(q\) are real numbers, has a complex root \(\alpha = 1 + \mathrm { i } \sqrt { 2 }\).

1. Write down a second complex root, \(\beta\).
2. Determine the third root, \(\gamma\).
3. Find the value of \(p\) and the value of \(q\).
4. Show that if \(n\) is an integer then \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }\) where \(\tan \theta = \sqrt { 2 }\).

2 The equation \(3 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\alpha\) and \(\beta\).
Find an equation with integer coefficients whose roots are \(3 - 2 \alpha\) and \(3 - 2 \beta\).

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.
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1. The equation \(5 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\frac { 1 } { p }\) and \(\frac { 1 } { q }\)
   1. Without solving the equation,
      1. show that \(p q = \frac { 5 } { 2 }\)
      2. determine the value of \(p + q\)
   2. Hence, without finding the values of \(p\) and \(q\), determine a quadratic equation with roots
   $$\frac { p } { p ^ { 2 } + 1 } \text { and } \frac { q } { q ^ { 2 } + 1 }$$ giving your answer in the form \(a x ^ { 2 } + b x + c = 0\) where \(a , b\) and \(c\) are integers.

1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).

2 The cubic equation \(x ^ { 3 } - 6 x ^ { 2 } + k x + 10 = 0\) has roots \(p - q , p\) and \(p + q\), where \(q\) is positive.

1. By considering the sum of the roots, find \(p\).
2. Hence, by considering the product of the roots, find \(q\).
3. Find the value of \(k\).

9. $$\mathrm { f } ( z ) = z ^ { 3 } - 8 z ^ { 2 } + p z - 24$$ 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 \(p\).
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2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\)
Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
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\(p - p - q - q\)

2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\)
Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
[0pt] [1 mark]
\(p - p - q - q\)

5 In the equation $$x ^ { 3 } + a x ^ { 2 } + b x + c = 0$$ the coefficients \(a , b\) and \(c\) are real. It is given that all the roots are real and greater than 1 .

1. Prove that \(a < - 3\).
2. By considering the sum of the squares of the roots, prove that \(a ^ { 2 } > 2 b + 3\).
3. By considering the sum of the cubes of the roots, prove that \(a ^ { 3 } < - 9 b - 3 c - 3\).

4 Use the substitution \(x = u + 2\) to find the exact value of the real root of the equation $$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$

1. The cubic equation

$$x ^ { 3 } + 3 x ^ { 2 } - 8 x + 6 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \(( \alpha - 1 ) , ( \beta - 1 )\) and \(( \gamma - 1 )\), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.
(5)

6 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots \(\alpha , \beta , \gamma\). Use the relation \(x = \sqrt { } y\) to show that the equation $$y ^ { 3 } + 2 y ^ { 2 } + y - 1 = 0$$ has roots \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\). Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).

1. Write down the value of \(S _ { 2 }\) and show that \(S _ { 4 } = 2\).
2. Find the values of \(S _ { 6 }\) and \(S _ { 8 }\).

8

1. Find the other roots.
2. Determine the value of \(m\).

1. $$\mathrm { f } ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.

2．（a）Show that \(( x + 1 )\) is a factor of \(2 x ^ { 3 } + 3 x ^ { 2 } - 1\)
（b）Solve the equation
（b）Solve the equation $$\sqrt { x ^ { 2 } + 2 x + 5 } = x + \sqrt { 2 x + 3 }$$

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 \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) 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 \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).

3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).

3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).

5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 . \end{aligned}$$ Write down the value of \(p\) and find the value of \(q\). Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).

3 Find a cubic equation with roots \(\alpha , \beta\) and \(\gamma\), given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.

3 Find a cubic equation with roots \(\alpha , \beta\) and \(\gamma\), given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.

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 roots of the quadratic equation \(x ^ { 2 } + x - 8 = 0\) are \(p\) and \(q\). Find the value of \(p + q + \frac { 1 } { p } + \frac { 1 } { q }\).

1. Find the values of \(\alpha , \beta\), and \(\gamma\).
2. Find the cubic equation with roots \(3 \alpha , 3 \beta , 3 \gamma\). 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.

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

1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).

3 The equation $$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$ has roots \(\alpha , \beta , \gamma\). Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\). Hence show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{array} \right)\) is singular.