This comprehensive guide is designed to enhance the understanding and skills of CIE IGCSE students in expanding algebraic expressions, focusing on the application of the distribution law to expand expressions involving products. Each example and practice problem is meticulously worked through with detailed equations to ensure mathematical accuracy and clarity.

Understanding the Distribution Law

The distribution law, or distributive property, is fundamental in algebra for expanding expressions. It is represented as $a(b + c) = ab + ac$, allowing for the multiplication of a sum by distributing the multiplier to each term within the parentheses.

Image courtesy of Online Math Learning

Worked Examples

Example 1: Expanding a Simple Product

To expand $3x(2x - 4y)$:

$3x \times 2x = 6x^2$

$3x \times (-4y) = -12xy$

Resulting in:

Example 2: Expanding a More Complex Expression

For $5y(3y^2 + 2y - 7)$:

$5y \times 3y^2 = 15y^3$

$5y \times 2y = 10y^2$

$5y \times (-7) = -35y$

Combining gives:

Example 3: Expanding Double Binomials

Expanding $(x + 3)(x - 2)$ involves:

Simplified to:

Example 4: Expanding with Coefficients

For $2(a + b)(2a - 3b)$, the detailed expansion is:

Example 5: Expanding Triple Terms

The expression $(2x - 3)(x^2 + x + 1)$ expands to:

Expanding Expressions with Multiple Terms

Expanding algebraic expressions with multiple terms can become quite intricate. The examples below illustrate how to approach the expansion of expressions involving trinomials and binomials, showcasing the versatility of the distribution law in algebra.

Example 6: Expanding a Trinomial by a Binomial

Consider expanding the expression $(x^2 + 2x + 1)(x - 3)$:

This involves distributing each term in the trinomial across the binomial:

Simplifying, we get:

Example 7: Expanding Two Trinomials

Expanding the expression $(2x^2 - x + 3)(x^2 + x - 1)$ involves a more complex process, given both multiplicands are trinomials:

The detailed expansion results in:

Upon combining like terms and simplifying, the final expression is: