Further Algebra：練習問題

The quartic equation \(2x^4 - 3x^3 + x^2 - 5x + 6 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).
Calculate the exact value of \(\sum \alpha^2 \beta\).
Hint: Use the identity \((\sum \alpha)(\sum \alpha\beta) = \sum \alpha^2\beta + 3\sum \alpha\beta\gamma\).

The equation \(x^3 - 2x^2 + 5x - 3 = 0\) has roots \(\alpha\), \(\beta\), and \(\gamma\).
Find the equation whose roots are \(2\alpha\), \(2\beta\), and \(2\gamma\), using the substitution \(y = 2x\).

Consider the rational function \(g(x) = \frac{x^3 + 2x^2 - x + 4}{x^2 + 1}\).
By first performing algebraic division, express \(g(x)\) in the form \(Ax + B + \frac{Cx + D}{x^2 + 1}\) and identify the resulting partial fraction components.

Consider the cubic equation \(2x^3 - 5x^2 + 4x - 7 = 0\) with roots \(\alpha\), \(\beta\), and \(\gamma\).
What is the value of the symmetric sum \(\alpha + \beta + \nu\)?

Express the rational function \(f(x) = \frac{2x^2 + 5x + 7}{(x-1)(x^2 + 4)}\) in partial fractions.

Let \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) be the roots of the quartic equation \(x^4 - 4x^3 + 2x^2 - 5x + 3 = 0\). Calculate the exact value of \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\).

The roots of the cubic equation \(2x^3 - 3x^2 + x - 5 = 0\) are \(\alpha, \beta, \gamma\).
By using the substitution \(y = x + 3\), find a cubic equation in \(y\) whose roots are \(\alpha+3, \beta+3, \gamma+3\). Give your answer in the form \(ay^3 + by^2 + cy + d = 0\) where \(a, b, c, d\) are integers.

Express the rational function \(\frac{3x^2 - x + 2}{(x - 1)(x^2 + 1)}\) in partial fractions of the form \(\frac{A}{x-1} + \frac{Bx+C}{x^2+1}\), where \(A\), \(B\), and \(C\) are constants to be determined.

The roots of the cubic equation \(x^3 + 5x^2 - 4x + 3 = 0\) are \(\alpha\), \(\beta\), and \(\gamma\).
(a) Write down the values of:
(i) \(\sum \alpha\)
(ii) \(\sum \alpha\beta\)
(iii) \(\alpha\beta\gamma\)

(b) Find a cubic equation with integer coefficients whose roots are \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\), and \(\frac{1}{\gamma}\).

(c) Calculate the value of \(\alpha^2 + \beta^2 + \gamma^2\).

Consider the rational function \(f(x) = \frac{2x^3 + 3x^2 + 7x + 5}{(x+1)(x^2 + 4)}\).
(a) Explain why the partial fraction decomposition of \(f(x)\) must include a constant term, and determine its value.

(b) Express \(f(x)\) in partial fractions in the form \(A + \frac{B}{x+1} + \frac{Cx + D}{x^2 + 4}\), where \(A, B, C,\) and \(D\) are constants to be determined.

(c) Using your result from part (b), find the exact value of \(\int_{0}^{2} f(x) \, dx\).