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

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

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

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

The cubic equation \(x^3 + bx^2 + cx - 1 = 0\), where \(b\) and \(c\) are constants, has roots \(\alpha, \beta, \gamma\).

It is given that the matrix \(\begin{pmatrix} 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{pmatrix}\) is singular.

(a) Show that \(\alpha^2 + \beta^2 + \gamma^2 = 3\).

(b) It is given that \(\alpha^3 + \beta^3 + \gamma^3 = 3\) and that the constants \(b\) and \(c\) are positive.

Find the values of \(b\) and \(c\).

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

(a) State, in terms of \(p\), the value of \(\alpha\beta + \beta\gamma + \gamma\alpha\).

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

(c) Deduce a cubic equation whose roots are \(\alpha\beta, \beta\gamma, \alpha\gamma\).

(d) Given that \(\alpha^2 + \beta^2 + \gamma^2 = \frac{1}{3}\), find the value of \(p\).

The cubic equation \(2x^3 + x^2 - px - 5 = 0\), 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 \(p\).

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

(a) Find a cubic equation whose roots are \(\alpha^2 + 1, \beta^2 + 1, \gamma^2 + 1\).

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

(c) Find the value of \((\alpha^2 + 1)^3 + (\beta^2 + 1)^3 + (\gamma^2 + 1)^3\).

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

(a) Find a cubic equation whose roots are \(\frac{1}{\alpha^3}, \frac{1}{\beta^3}, \frac{1}{\gamma^3}\).

(b) Find the value of \(\frac{1}{\alpha^6} + \frac{1}{\beta^6} + \frac{1}{\gamma^6}\).

(c) Find also the value of \(\frac{1}{\alpha^9} + \frac{1}{\beta^9} + \frac{1}{\gamma^9}\).

The cubic equation \(x^3 + 5x^2 + 10x - 2 = 0\) has roots \(\alpha, \beta, \gamma\).

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

(b) Show that the matrix \(\begin{pmatrix} 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{pmatrix}\) is singular.

The cubic equation \(x^3 + bx^2 + d = 0\) has roots \(\alpha, \beta, \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).

(a) Show that \(4b^3 + 27d = 0\).

(b) Given that \(2\alpha^2 + \gamma^2 = 3b\), find the values of \(b\) and \(d\).

The cubic equation \(2x^3 - 4x^2 + 3 = 0\) has roots \(\alpha, \beta, \gamma\). Let \(S_n = \alpha^n + \beta^n + \gamma^n\).

1. (a) State the value of \(S_1\) and find the value of \(S_2\).
2. (b)
   1. Express \(S_{n+3}\) in terms of \(S_{n+2}\) and \(S_n\).
   2. Hence, or otherwise, find the value of \(S_4\).
3. (c) Use the substitution \(y = S_1 - x\), 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. (d) Find the value of \(\frac{1}{\alpha + \beta} + \frac{1}{\beta + \gamma} + \frac{1}{\gamma + \alpha}\).

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 \(x^3 + bx^2 + cx + d = 0\) has roots \(\alpha, \beta, \gamma\).

Find the values of \(b, c\) and \(d\).

3 The cubic equation \(x^{3}+c x+1=0\), where \(c\) is a constant, has roots \(\alpha, \beta, \gamma\).
(a) Find a cubic equation whose roots are \(\alpha^{3}, \beta^{3}, \gamma^{3}\).

(b) Show that \(\alpha^{6}+\beta^{6}+\gamma^{6}=3-2 c^{3}\).

(c) Find the real value of \(c\) for which the matrix \(\left(\begin{array}{ccc}1 & \alpha^{3} & \beta^{3} \\ \alpha^{3} & 1 & \gamma^{3} \\ \beta^{3} & \gamma^{3} & 1\end{array}\right)\) is singular.

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

(b) It is given that \(\alpha^{2}+\beta^{2}+\gamma^{2}=2(\alpha+\beta+\gamma)\).
(i) Find the value of \(p\).

(ii) Find the value of \(\alpha^{3}+\beta^{3}+\gamma^{3}\).

6 The equation
\(x^{3}-x+1=0\)
has roots \(\alpha, \beta, \gamma\).
(i) Use the relation \(x=y^{\frac{1}{3}}\) to show that the equation
\(y^{3}+3 y^{2}+2 y+1=0\)
has roots \(\alpha^{3}, \beta^{3}, \gamma^{3}\). Hence write down the value of \(\alpha^{3}+\beta^{3}+\gamma^{3}\).

Let \(S_{n}=\alpha^{n}+\beta^{n}+\gamma^{n}\).
(ii) Find the value of \(S_{-3}\).

(iii) Show that \(S_{6}=5\) and find the value of \(S_{9}\).

9 A cubic equation \(x^{3}+b x^{2}+c x+d=0\) has real roots \(\alpha, \beta\) and \(\gamma\) such that
\(\begin{aligned}
\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} & =-\frac{5}{12}, \\
\alpha \beta \gamma & =-12, \\
\alpha^{3}+\beta^{3}+\gamma^{3} & =90 .
\end{aligned}\)
(i) Find the values of \(c\) and \(d\).

(ii) Express \(\alpha^{2}+\beta^{2}+\gamma^{2}\) in terms of \(b\).

(iii) Show that \(b^{3}-15 b+126=0\).

(iv) Given that \(3+\mathrm{i} \sqrt{ }(12)\) is a root of \(y^{3}-15 y+126=0\), deduce the value of \(b\).

The equation \(x^{3}+2 x^{2}+x+7=0\) has roots \(\alpha, \beta, \gamma\).
(i) Use the relation \(x^{2}=-7 y\) to show that the equation
\(49 y^{3}+14 y^{2}-27 y+7=0\)
has roots \(\frac{\alpha}{\beta \gamma}, \frac{\beta}{\gamma \alpha}, \frac{\gamma}{\alpha \beta}\).

(ii) Show that \(\frac{\alpha^{2}}{\beta^{2} \gamma^{2}}+\frac{\beta^{2}}{\gamma^{2} \alpha^{2}}+\frac{\gamma^{2}}{\alpha^{2} \beta^{2}}=\frac{58}{49}\).

(iii) Find the exact value of \(\frac{\alpha^{3}}{\beta^{3} \gamma^{3}}+\frac{\beta^{3}}{\gamma^{3} \alpha^{3}}+\frac{\gamma^{3}}{\alpha^{3} \beta^{3}}\).

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 \(a r^{-1}\).
(i) Find the numerical values of the roots.

(ii) Deduce the value of \(k\).

The equation
\(9 x^{3}-9 x^{2}+x-2=0\)
has roots \(\alpha, \beta, \gamma\).
(i) Use the substitution \(y=3 x-1\) to show that \(3 \alpha-1,3 \beta-1,3 \gamma-1\) are the roots of the equation
\(y^{3}-2 y-7=0 .\)

The sum \((3 \alpha-1)^{n}+(3 \beta-1)^{n}+(3 \gamma-1)^{n}\) is denoted by \(S_{n}\).
(ii) Find the value of \(S_{3}\).

~~。
(iii) Find the value of \(S_{-2}\).

The roots of the equation
\(x^{3}+p x^{2}+q x+r=0\)
are \(\alpha, 2 \alpha, 4 \alpha\), where \(p, q, r\) and \(\alpha\) are non-zero real constants.
(i) Show that
\(2 p \alpha+q=0 .\)

(ii) Show that
\(p^{3} r-q^{3}=0 .\)

(a) The cubic \(x^3+bx^2+cx+d=0\) has roots \(\alpha,\beta,\gamma\). Given \(\alpha+\beta+\gamma=2\), \(\alpha^2+\beta^2+\gamma^2=3\), and \(\alpha^4+\beta^4+\gamma^4=5\), find \(b\) and \(c\).

(b) Find the value of \(d\).

The roots of the cubic equation
\(x^{3}-5 x^{2}+13 x-4=0\)
are \(\alpha, \beta, \gamma\).
(i) Find the value of \(\alpha^{2}+\beta^{2}+\gamma^{2}\).

(ii) Find the value of \(\alpha^{3}+\beta^{3}+\gamma^{3}\).

By finding a cubic equation whose roots are \(\alpha, \beta\) and \(\gamma\), solve the set of simultaneous equations
\(\begin{aligned} \alpha+\beta+\gamma & =-1 \\ \alpha^{2}+\beta^{2}+\gamma^{2} & =29 \\ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} & =-1 \end{aligned}\)

The roots of the cubic equation \(x^{3}+2 x^{2}-3=0\) are \(\alpha, \beta\) and \(\gamma\).
(i) By using the substitution \(y=\frac{1}{x^{2}}\), find the cubic equation with roots \(\frac{1}{\alpha^{2}}, \frac{1}{\beta^{2}}\) and \(\frac{1}{\gamma^{2}}\).

(ii) Hence find the value of \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}\).

(iii) Find also the value of \(\frac{1}{\alpha^{2} \beta^{2}}+\frac{1}{\beta^{2} \gamma^{2}}+\frac{1}{\gamma^{2} \alpha^{2}}\).

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 \(p+q^{2}=1\).

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 \(r\).

The roots of the cubic equation \(x^{3}-7 x^{2}+2 x-3=0\) are \(\alpha, \beta\) and \(\gamma\). Find the values of
(i) \(\frac{1}{(\alpha \beta)(\beta \gamma)(\gamma \alpha)}\),
(ii) \(\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}\),
(iii) \(\frac{1}{\alpha^{2} \beta \gamma}+\frac{1}{\alpha \beta^{2} \gamma}+\frac{1}{\alpha \beta \gamma^{2}}\).

Deduce a cubic equation, with integer coefficients, having roots \(\frac{1}{\alpha \beta}, \frac{1}{\beta \gamma}\) and \(\frac{1}{\gamma \alpha}\).

Find the cubic equation with roots \(\alpha, \beta\) and \(\gamma\) such that
\(\begin{aligned} \alpha+\beta+\gamma & =3 \\ \alpha^{2}+\beta^{2}+\gamma^{2} & =1 \\ \alpha^{3}+\beta^{3}+\gamma^{3} & =-30 \end{aligned}\)
giving your answer in the form \(x^{3}+p x^{2}+q x+r=0\), where \(p, q\) and \(r\) are integers to be found.

The cubic equation
\(z^{3}-z^{2}-z-5=0\)
has roots \(\alpha, \beta\) and \(\gamma\). Show that the value of \(\alpha^{3}+\beta^{3}+\gamma^{3}\) is 19 .

Find the value of \(\alpha^{4}+\beta^{4}+\gamma^{4}\).

Show that the cubic equation with roots \(\frac{\alpha-1}{\alpha}, \frac{\beta-1}{\beta}\) and \(\frac{\gamma-1}{\gamma}\) may be found using the substitution \(z=\frac{1}{1-x}\), and find this equation, 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.

The roots of the cubic equation \(2 x^{3}+x^{2}-7=0\) are \(\alpha, \beta\) and \(\gamma\). Using the substitution \(y=1+\frac{1}{x}\), or otherwise, find the cubic equation whose roots are \(1+\frac{1}{\alpha}, 1+\frac{1}{\beta}\) and \(1+\frac{1}{\gamma}\), giving your answer in the form \(a y^{3}+b y^{2}+c y+d=0\), where \(a, b, c\) and \(d\) are constants to be found.

The cubic equation \(2 x^{3}-3 x^{2}+4 x-10=0\) has roots \(\alpha, \beta\) and \(\gamma\).
(i) Find the value of \((\alpha+1)(\beta+1)(\gamma+1)\).

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

Answer only one of the following two alternatives.

EITHER
The roots of the quartic equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\) are \(\alpha, \beta, \gamma\) and \(\delta\). Find the values of
(i) \(\alpha+\beta+\gamma+\delta\),

(ii) \(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\),

(iii) \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}\),
(iv) \(\frac{\alpha}{\beta \gamma \delta}+\frac{\beta}{\alpha \gamma \delta}+\frac{\gamma}{\alpha \beta \delta}+\frac{\delta}{\alpha \beta \gamma}\).

Using the substitution \(y=x+1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\).

OR

The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that if \(\mathbf{A}\) is non-singular then
(i) \(\lambda \neq 0\),

(ii) the matrix \(\mathbf{A}^{-1}\) has \(\lambda^{-1}\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector.

The \(3 \times 3\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 2 & -4 \\ 0 & -1 & 5 \\ 0 & 0 & 3 \end{array}\right) \quad \text { and } \quad \mathbf{B}=(\mathbf{A}+3 \mathbf{I})^{-1}\)
where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf{P}\), and a diagonal matrix \(\mathbf{D}\), such that \(\mathbf{B}=\mathbf{P D P}^{-1}\).

Answer only one of the following two alternatives.

EITHER
The roots of the quartic equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\) are \(\alpha, \beta, \gamma\) and \(\delta\). Find the values of
(i) \(\alpha+\beta+\gamma+\delta\),

(ii) \(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\),

(iii) \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}\),
(iv) \(\frac{\alpha}{\beta \gamma \delta}+\frac{\beta}{\alpha \gamma \delta}+\frac{\gamma}{\alpha \beta \delta}+\frac{\delta}{\alpha \beta \gamma}\).

Using the substitution \(y=x+1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\).

OR

The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that if \(\mathbf{A}\) is non-singular then
(i) \(\lambda \neq 0\),

(ii) the matrix \(\mathbf{A}^{-1}\) has \(\lambda^{-1}\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector.

The \(3 \times 3\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 2 & -4 \\ 0 & -1 & 5 \\ 0 & 0 & 3 \end{array}\right) \quad \text { and } \quad \mathbf{B}=(\mathbf{A}+3 \mathbf{I})^{-1}\)
where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf{P}\), and a diagonal matrix \(\mathbf{D}\), such that \(\mathbf{B}=\mathbf{P D P}^{-1}\).

The cubic equation \(x^{3}-2 x^{2}-3 x+4=0\) has roots \(\alpha, \beta, \gamma\). Given that \(c=\alpha+\beta+\gamma\), state the value of \(c\).

Use the substitution \(y=c-x\) to find a cubic equation whose roots are \(\alpha+\beta, \beta+\gamma, \gamma+\alpha\).

Find a cubic equation whose roots are \(\frac{1}{\alpha+\beta}, \frac{1}{\beta+\gamma}, \frac{1}{\gamma+\alpha}\).

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

The roots of the equation \(x^{4}-4 x^{2}+3 x-2=0\) are \(\alpha, \beta, \gamma\) and \(\delta\); the sum \(\alpha^{n}+\beta^{n}+\gamma^{n}+\delta^{n}\) is denoted by \(S_{n}\). By using the relation \(y=x^{2}\), or otherwise, show that \(\alpha^{2}, \beta^{2}, \gamma^{2}\) and \(\delta^{2}\) are the roots of the equation
\(y^{4}-8 y^{3}+12 y^{2}+7 y+4=0\)

State the value of \(S_{2}\) and hence show that
\(S_{8}=8 S_{6}-12 S_{4}-72 .\)

The cubic equation \(x^{3}-p x-q=0\), where \(p\) and \(q\) are constants, has roots \(\alpha, \beta, \gamma\). Show that
(i) \(\alpha^{2}+\beta^{2}+\gamma^{2}=2 p\),

(ii) \(\alpha^{3}+\beta^{3}+\gamma^{3}=3 q\),

(iii) \(6\left(\alpha^{5}+\beta^{5}+\gamma^{5}\right)=5\left(\alpha^{3}+\beta^{3}+\gamma^{3}\right)\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\).

Answer only one of the following two alternatives.

EITHER

State the fifth roots of unity in the form \(\cos \theta+\mathrm{i} \sin \theta\), where \(-\pi\lt \theta \leqslant \pi\).

Simplify
\(\left(x-\left[\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right]\right)\left(x-\left[\cos \frac{2}{5} \pi-i \sin \frac{2}{5} \pi\right]\right) .\)

Hence find the real factors of
\(x^{5}-1\)

Express the six roots of the equation
\(x^{6}-x^{3}+1=0\)
as three conjugate pairs, in the form \(\cos \theta \pm \mathrm{i} \sin \theta\).

Hence find the real factors of
\(x^{6}-x^{3}+1\)

OR

Given that
\(y^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}-6 y^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+3 y^{3}=25 \mathrm{e}^{-2 x}\)
and that \(v=y^{3}\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}-6 \frac{\mathrm{~d} v}{\mathrm{~d} x}+9 v=75 \mathrm{e}^{-2 x}\)

Find the particular solution for \(y\) in terms of \(x\), given that when \(x=0, y=2\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=1\).

The cubic equation \(x^{3}-p x-q=0\), where \(p\) and \(q\) are constants, has roots \(\alpha, \beta, \gamma\). Show that
(i) \(\alpha^{2}+\beta^{2}+\gamma^{2}=2 p\),

(ii) \(\alpha^{3}+\beta^{3}+\gamma^{3}=3 q\),

(iii) \(6\left(\alpha^{5}+\beta^{5}+\gamma^{5}\right)=5\left(\alpha^{3}+\beta^{3}+\gamma^{3}\right)\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\).

Answer only one of the following two alternatives.

EITHER

State the fifth roots of unity in the form \(\cos \theta+\mathrm{i} \sin \theta\), where \(-\pi\lt \theta \leqslant \pi\).

Simplify
\(\left(x-\left[\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right]\right)\left(x-\left[\cos \frac{2}{5} \pi-i \sin \frac{2}{5} \pi\right]\right) .\)

Hence find the real factors of
\(x^{5}-1\)

Express the six roots of the equation
\(x^{6}-x^{3}+1=0\)
as three conjugate pairs, in the form \(\cos \theta \pm \mathrm{i} \sin \theta\).

Hence find the real factors of
\(x^{6}-x^{3}+1\)

OR
Given that
\(y^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}-6 y^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+3 y^{3}=25 \mathrm{e}^{-2 x}\)
and that \(v=y^{3}\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}-6 \frac{\mathrm{~d} v}{\mathrm{~d} x}+9 v=75 \mathrm{e}^{-2 x}\)

Find the particular solution for \(y\) in terms of \(x\), given that when \(x=0, y=2\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=1\).

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 \(k\).

The equation
\(x^{3}+x-1=0\)
has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(\alpha^{3}, \beta^{3}, \gamma^{3}\) is
\(y^{3}-3 y^{2}+4 y-1=0 .\)

Hence find the value of \(\alpha^{6}+\beta^{6}+\gamma^{6}\).

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 \(r p^{3}=q^{3}\).

The cubic equation \(x^{3}-x^{2}-3 x-10=0\) has roots \(\alpha, \beta, \gamma\).
(i) Let \(u=-\alpha+\beta+\gamma\). Show that \(u+2 \alpha=1\), and hence find a cubic equation having roots \(-\alpha+\beta+\gamma\), \(\alpha-\beta+\gamma, \alpha+\beta-\gamma\).
(ii) State the value of \(\alpha \beta \gamma\) and hence find a cubic equation having roots \(\frac{1}{\beta \gamma}, \frac{1}{\gamma \alpha}, \frac{1}{\alpha \beta}\).

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 \(p+q^{2}=1\).

A cubic equation has roots \(\alpha, \beta\) and \(\gamma\) such that
\(\begin{aligned} \alpha+\beta+\gamma & =4, \\ \alpha^{2}+\beta^{2}+\gamma^{2} & =14, \\ \alpha^{3}+\beta^{3}+\gamma^{3} & =34 . \end{aligned}\)

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

Show that the cubic equation is
\(x^{3}-4 x^{2}+x+6=0\)
and solve this equation.

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 \(S_{n}=\alpha^{n}+\beta^{n}+\gamma^{n}\).
(i) Write down the value of \(S_{2}\) and show that \(S_{4}=2\).

(ii) Find the values of \(S_{6}\) and \(S_{8}\).

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.

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

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 \(\left(\begin{array}{ccc}1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1\end{array}\right)\) is singular.

The equation
\(x^{4}-x^{3}-1=0\)
has roots \(\alpha, \beta, \gamma, \delta\). By using the substitution \(y=x^{3}\), or by any other method, find the exact valu. \(\alpha^{6}+\beta^{6}+\gamma^{6}+\delta^{6}\).

Use de Moivre's theorem to show that
\(\sin 5 \theta=16 \sin ^{5} \theta-20 \sin ^{3} \theta+5 \sin \theta .\)

Hence find all the roots of the equation
\(32 x^{5}-40 x^{3}+10 x+1=0\)
in the form \(\sin (q \pi)\), where \(q\) is a positive rational number.

The equation
\(x^{4}+x^{3}+c x^{2}+4 x-2=0,\)
where \(c\) is a constant, has roots \(\alpha, \beta, \gamma, \delta\).
(i) Use the substitution \(y=\frac{1}{x}\) to find an equation which has roots \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}, \frac{1}{\delta}\).

(ii) Find, in terms of \(c\), the values of \(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\) and \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}+\frac{1}{\delta^{2}}\).

(iii) Hence find
\(\left(\alpha-\frac{1}{\alpha}\right)^{2}+\left(\beta-\frac{1}{\beta}\right)^{2}+\left(\gamma-\frac{1}{\gamma}\right)^{2}+\left(\delta-\frac{1}{\delta}\right)^{2}\)
in terms of \(c\).

(iv) Deduce that when \(c=-3\) the roots of the given equation are not all real.

The equation
\(x^{3}+5 x+3=0\)
has roots \(\alpha, \beta, \gamma\). Use the substitution \(x=-\frac{3}{y}\) to find a cubic equation in \(y\) and show that the roots of this equation are \(\beta \gamma, \gamma \alpha, \alpha \beta\).

Find the exact values of \(\beta^{2} \gamma^{2}+\gamma^{2} \alpha^{2}+\alpha^{2} \beta^{2}\) and \(\beta^{3} \gamma^{3}+\gamma^{3} \alpha^{3}+\alpha^{3} \beta^{3}\).