Notation, vocabulary and manipulation：練習問題

Simplify the expression by collecting like terms:
\( 4x + 3y - 2x + 5y \)

Factorise the following quadratic expression completely:
\( x^2 - 9x + 20 \)

Expand and simplify the product of three binomials:
\( (x + 1)(x + 2)(x + 3) \)

Which of the following represents the expression for "the square of the sum of a and b"?

Expand and simplify the following expression:
\( 3(2x - 4) + 2(x + 5) \)

Write down the algebraic expression for the following statement:
The sum of three lots of \( x \) and five lots of \( y \).

Expand the brackets and simplify the following expression by collecting like terms:
\( 5(2x - 3) - 3(x - 4) \)

Find the value of the expression \( 3a^2 - 4b \) when \( a = 4 \) and \( b = 5 \).

An algebraic identity is given as \( 3(2x + 4) \equiv ax + b \).

(a) Find the values of the constants \( a \) and \( b \).

(b) Explain the difference between an equation and an identity using this example.

Simplify the following algebraic expressions as much as possible:

(a) \( \frac{18x^4y^3}{6x^2y} \)
(b) \( (2a^2b)^3 \)
(c) Factorise completely the expression \( 15p^2q - 10pq^2 \).