代數與函數：練習題

Rationalize the denominator of \(\frac{4}{\sqrt{5}-1}\) and simplify the expression completely.

Find the set of values of \(x\) for which \(2x^2 + x > 15\).

Given that \(x^3 - 2x^2 - 5x + 6 = (x-1)(ax^2 + bx + c)\), find the values of the constants \(a, b, \) and \(c\) and hence solve the equation \(x^3 - 2x^2 - 5x + 6 = 0\).

Simplify the expression completely:
\( (27x^6)^{\frac{1}{3}} \times (4x^{-2})^{\frac{1}{2}} \)

Solve the inequality \(\frac{4}{x - 2} < x + 1\).

Find the exact value of \( \left( \frac{8}{27} \right)^{-\frac{2}{3}} \).

Find the value of the discriminant for the quadratic function \( f(x) = 3x^2 - 4x + 2 \) and use it to determine the number of real roots of the equation \( f(x) = 0 \).

The function \( f \) is defined by \( f(x) = e^{2x} + 3 \) for \( x \in \mathbb{R} \). Find an expression for \( f^{-1}(x) \) and determine its domain.

The polynomial \( f(x) \) is defined by \( f(x) = x^3 + ax^2 - 7x + b \), where \( a \) and \( b \) are constants.
Given that \( (x - 1) \) is a factor of \( f(x) \) and that when \( f(x) \) is divided by \( (x + 2) \), the remainder is 12.

Part (a)

Find the value of \( a \) and the value of \( b \).

Part (b)

Factorise \( f(x) \) completely.

A function \( f \) is defined by \( f(x) = x^2 + 6x + 5 \) for \( x \in \mathbb{R} \).

Part (a)
Express \( f(x) \) in the form \( (x + a)^2 + b \), where \( a \) and \( b \) are constants to be found.

Part (b)
State the coordinates of the vertex of the graph of \( y = f(x) \).

Part (c)
The curve \( y = f(x) \) is transformed to the curve \( y = g(x) \) by a translation of \( \begin{pmatrix} 2 \\ -3 \end{pmatrix} \). Find the equation of \( g(x) \) in the form \( g(x) = x^2 + px + q \).