It is known that two of the roots of the equation: \(2{x^3} - 2{x^2} - kx + 6 = 0\) are equal in value but opposite in sign. Find the value of \(k\).

\(P(x) = {x^3} - 6{x^2} + 11x - 6\) has one of the roots equal to the sum of the other two roots. The values of the three roots are?

If \(\alpha \) and \(\beta \) are the roots of the equation \({x^2} + 4x - 2 = 0\), then the quadratic equation whose roots are \(\dfrac{\alpha }{\beta }\) and \(\dfrac{\beta }{\alpha }\) is?

The roots of \({x^3} - 6{x^2} + 3x + m = 0\) are in arithmetic progression. The value of \(m\) is?

If \(\alpha \) and \(\beta \) are the roots of the  \({x^2} + 8x - 5 = 0\),find the quadratic equation whose roots are \(\dfrac{\alpha }{\beta }\) and \(\dfrac{\beta }{a}\).

One of the roots of \(P(x) = {x^3} + p{x^2} + 1\) is equal to the sum of the other two roots of \(P(x)\).Find the value of \(p\).

The polynomial \(P(x) = {x^3} - {x^2} + 4x - 6\) has roots \(\alpha \),\(\beta \) and \(\gamma \).

Evaluate:

\(i)\,\alpha + \beta + \gamma \)

\(ii)\,\alpha \beta \gamma \)

\(iii)\,\dfrac{1}{{\alpha \beta }} + \dfrac{1}{{\beta \gamma }} + \dfrac{1}{{\alpha \gamma }}\)

The polynomial \(P(x)\) is given by \(P(x) = p{x^3} - 8{x^2} + qx - 32\), where \(p\) and \(q\) are constants. The three zeros of \(P(x)\) are \(-2,4\) and \(\alpha\). Find the value of \(\alpha\).

The polynomial \(P(x) = 10x^3 - 39x^2 + ax+b = 0\) has a zero at \(x = 1\) and the remaining zeros are reciprocals of each other.

i) Find the values of \(a\) and \(b\).

ii) Find all the zeros of the polynomial.

The polynomial \(P(x) = 9x^4 + 9x^3 - 25x^2 - 27x -6\) has the roots \(\alpha,\, \beta,\,\gamma,\, \delta\). Given the condition that \(\alpha + \beta = 0\),

i) Show that \(\gamma + \delta = -1\)
ii) By considering the sum of the roots three at a time show that \(-\beta^2(\gamma + \delta) = 3\)
iii) By considering the sum of the roots two at a time show that \(-\beta^2 + \gamma\delta = \dfrac{-25}{9}\)
iv) Hence or otherwise determine the roots of \(P(x)\)

Solve the equation \(16x^3 - 26x^2 - 13x + 2 = 0\) given that the roots are in geometric sequence.

The polynomial \(P(x) = x^4 + 4x^3 + x^2 -6x + 2\) has roots; \(\alpha,\, \beta,\, \gamma,\, \delta\). Given the condition that \(\alpha\beta = -1\):

i) Show that \(\gamma\delta = -2\).

ii) By considering the sum of the roots show that \(\gamma + \delta = -4-(\alpha+\beta)\).

iii) By considering the sum of the roots three at a time show that \(\gamma + \delta = -6-2(\alpha+\beta)\).

iv) Hence or otherwise determine the roots of \(P(x)\)

The \(\alpha, \beta, \delta\) are the roots of \(x^3 -x+1 = 0\) find the value of \[\left(\alpha + \frac{1}{\beta} \right)\left(\beta + \frac{1}{\delta} \right)\left(\delta + \frac{1}{\alpha} \right)\]

The polynomial \(P(x) = 8x^3 - 12x^2 -2x+3\) has roots forming consecutive terms of an arithmetic sequence. Find these roots.

The polynomial \(P(x) = 24x^3 - 38x^2 + 19x - 3\) has roots forming consecutive terms of a geometric sequence. Find these roots.

If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 + mx+n = 0\), find the roots of \(nx^2-mx+1 = 0\) in terms of \(\alpha\) and \(\beta\).

Express \(\cos 3\theta = -\frac{1}{2}\) in terms of \(\cos \theta\) and by letting \(x = \cos\theta\) show that \(\cos 3\theta = -\frac{1}{2}\) can be expressed as

\[8x^3-6x+1=0\]

Hence deduce that:

i) \(\displaystyle \cos \frac{\pi}{9} = \cos \frac{2\pi}{9} + \cos\frac{4\pi}{9}\)

ii) \(\displaystyle \sec\frac{\pi}{9}\sec\frac{2\pi}{9}\sec\frac{4\pi}{9} = 8\)

iii) \(\displaystyle \tan^2\frac{\pi}{9}+\tan^2\frac{2\pi}{9}+\tan^2\frac{4\pi}{9} = 33\)

NOTE: This is a very challenging and hard question

i) Prove that \( \displaystyle \tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1-3\tan^2\theta}\)

ii) Let \(x =\tan\theta\), hence find the three roots of \( \displaystyle x^3-3x^2-3x+1=0\)

iii) Find the value of

\((\alpha)\quad\displaystyle \tan \frac{\pi}{12}+\tan\frac{5\pi}{12}\)

\((\beta)\quad\displaystyle \tan\frac{\pi}{12}\times\tan\frac{5\pi}{12}\)

Note: this is a very hard and challenging question