Algebra and functions：練習問題

Find the values of \(x\) for which the inequality \(3x - 7 > 5x + 3\) is satisfied. Express the result in set notation.

The polynomial \(f(x) = x^3 - 4x^2 + ax + 6\) has a factor of \((x - 2)\). Find the value of the constant \(a\).

Given that \(f(x) = 2x - 3\) and \(g(x) = x^2 + 1\), find the composite function \(fg(x)\).

The polynomial \( P(x) = 2x^3 - 5x^2 - x + k \) has a factor \( (x - 2) \). Find the value of \( k \).

Simplify the following expression using the laws of indices:

\(\frac{(2x^2)^3}{4x^4}\)

Find the range of values of \( k \) for which the quadratic equation \( x^2 + 6x + k = 0 \) has two distinct real roots.

The curve \( y = x^2 \) is translated by the vector \( \begin{pmatrix} 0 \\ -4 \end{pmatrix} \) and then stretched parallel to the \( y \)-axis by a scale factor of 3. Find the equation of the resulting curve.

Rationalise the denominator of the expression \( \frac{1}{\sqrt{3} - 1} \) and simplify your answer fully.

Solve the simultaneous equations:

\( y - 2x = 3 \)
\( x^2 + xy = 10 \)

A curve has the equation \( y = 2x^3 - 9x^2 + 12x + 5 \).

(a) Find the coordinates of the two stationary points on the curve.
(b) By considering the second derivative, determine the nature of each stationary point.