代數運算與公式：練習題

Factorise the quadratic expression \(x^2 - 7x + 10\) completely.

Rearrange the formula \(y = \frac{2x - 5}{3 + x}\) to make \(x\) the subject.

The polynomial \(f(x) = 2x^3 + ax^2 - 5x - 6\) has \((x - 2)\) as a factor. Find the value of the constant \(a\) and hence find the other factors of \(f(x)\).

Simplify the algebraic expression \((3x^2y^3)^3\) using the laws of indices.

Solve the linear inequality \(3(2x - 1) \le 4x + 7\) and represent the solution.

Simplify the expression \( \frac{a^7 \times a^{-3}}{a^2} \), giving your answer in the form \( a^k \).

Using the factor theorem, show that \( (x - 2) \) is a factor of the cubic expression \( P(x) = 2x^3 - 3x^2 - 3x + 2 \) and hence find all the values of \( x \) such that \( P(x) = 0 \).

Solve the inequality \(5x - 2 \leq 13\) and give your answer in the form \(x \leq k\).

(a) Expand and simplify the expression \(5(2x - 3) - 3(x + 4)\).
(b) Solve the equation \(\frac{2y + 5}{3} = 7\).
(c) Factorise the quadratic expression \(x^2 - 9x + 20\).

Solve the following equation for \(x\):

\( \frac{4}{x - 2} - \frac{3}{x + 1} = 1 \)

Show every step of your working clearly and give your answers to 2 decimal places.