Roots with special relationships

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 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 \(k\).

5 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , - 3 \alpha\) and \(\alpha + 3\). Find the values of \(\alpha , p\) and \(q\).

3 The roots of the cubic equation \(4 x ^ { 3 } - 12 x ^ { 2 } + k x - 3 = 0\) may be written \(a - d , a\) and \(a + d\). Find the roots and the value of \(k\).

4 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } + p x + q = 0\) are \(2 w , - 6 w\) and \(3 w\). Find the values of the roots and the values of \(p\) and \(q\).

3 The cubic equation \(3 x ^ { 3 } + 8 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { \alpha } { 6 }\) and \(\alpha - 7\). Find the values of \(\alpha , p\) and \(q\).

2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\frac { \beta } { k } , \beta , k \beta\), where \(p , q , r , k\) and \(\beta\) are non-zero real constants. Show that \(\beta = - \frac { q } { p }\). Deduce that \(r p ^ { 3 } = q ^ { 3 }\).

1 The equation \(x ^ { 3 } + p x + q = 0\), where \(p\) and \(q\) are constants, with \(q \neq 0\), has one root which is the reciprocal of another root. Prove that \(p + q ^ { 2 } = 1\).

4 It is given that the equation $$x ^ { 3 } - 21 x ^ { 2 } + k x - 216 = 0$$ where \(k\) is a constant, has real roots \(a , a r\) and \(a r ^ { - 1 }\).

1. Find the numerical values of the roots.
2. Deduce the value of \(k\).

5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where \(k\) is a constant, has roots \(a - d , a , a + d\). Find the numerical values of the roots and determine the value of \(k\).

1 The quartic equation \(x ^ { 4 } - p x ^ { 2 } + q x - r = 0\), where \(p , q\) and \(r\) are real constants, has two pairs of equal roots. Show that \(p ^ { 2 } + 4 r = 0\) and state the value of \(q\).

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

5 The cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { 2 } { \alpha }\) and \(\alpha + \frac { 2 } { \alpha }\). Find

• the values of the roots of the equation,
• the value of \(p\).

10 The equation
\(4 x ^ { 4 } + 16 x ^ { 3 } + a x ^ { 2 } + b x + 6 = 0\),
where \(a\) and \(b\) are real, has roots \(\alpha , \frac { 2 } { \alpha } , \beta\) and \(3 \beta\).

1. Given that \(\beta < 0\), determine all 4 roots.
2. Determine the values of \(a\) and \(b\).

10 The equation \(\mathrm { x } ^ { 3 } - 4 \mathrm { x } ^ { 2 } + 7 \mathrm { x } + \mathrm { c } = 0\), where \(c\) is a constant, has roots \(\alpha , \beta\) and \(\alpha + \beta\).

1. Determine the roots of the equation.
2. Find c.

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 \(q\).

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.

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.
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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 \(p\).

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 \(p\).

8 In this question you must show detailed reasoning. The equation \(\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0\) has roots \(\alpha , \beta\) and \(\frac { \alpha } { \beta }\). Given that \(\alpha > 0\), find, in any order,

• the roots of the equation,
• the value of \(k\).

8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).

1. Evaluate, in exact form, the roots of the equation.
2. Find \(k\).

4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.

8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where \(k\) is a constant, form an arithmetic sequence. Find the roots of the equation.