代數：練習題

Solve the equation \( 8^{x-1} = 32 \) for the value of \( x \).

Solve the inequality \( x^2 - 5x - 6 < 0 \).

Express \(\frac{5x^2 - 2x + 11}{(x + 1)(x^2 + 4)}\) in the form \(\frac{A}{x + 1} + \frac{Bx + C}{x^2 + 4}\) and determine the value of \(B + C\).

Factorise the quadratic expression \( 4x^2 - 25 \) completely.

The polynomial \( f(x) = x^3 + ax + 6 \) leaves a remainder of 4 when divided by \( (x - 1) \). Find the value of the constant \( a \).

A curve has the equation \( y = x^2 - 10x + 25 \). Find the coordinates of the vertex of this curve.

Use the factor theorem to show that \((x + 2)\) is a factor of the polynomial \(P(x) = x^3 + 5x^2 + 2x - 8\), and hence find the remainder when \(P(x)\) is divided by \((x - 1)\).

By using a suitable substitution, solve the equation \( x - 5\sqrt{x} + 6 = 0 \).

(a) Express \( \sqrt{108} - \sqrt{48} \) in the form \( k\sqrt{3} \), where \( k \) is an integer.
(b) Solve the simultaneous equations:
\( y - 3x = 1 \)
\( y = x^2 - 2x + 5 \)
giving your answers as coordinates \( (x, y) \).

The polynomial \( f(x) = x^3 - 4x^2 + kx + 6 \) has a factor \( (x - 2) \).
(a) Use the factor theorem to find the value of the constant \( k \).
(b) With this value of \( k \), factorise \( f(x) \) completely into three linear factors.