Creating detailed and computation-focused study notes for the topic of factorization, especially tailored for CIE IGCSE students, involves presenting the material with a greater emphasis on mathematical expressions and less on textual explanations. Here, we'll delve deeper into factorization techniques with comprehensive examples and step-by-step solutions using mathematical notation.

**Introduction to Factorization**

Factorization is a key algebraic technique that simplifies expressions and equations by expressing them as the product of their factors. This method is crucial for solving quadratic equations, simplifying algebraic expressions, and solving higher-degree polynomial equations.

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**Greatest Common Factor (GCF)**

Identifying the GCF of the terms in an expression is the first step in factorization. The GCF is the highest number (and/or algebraic term) that divides all the coefficients (and/or terms) of the expression.

**Example: Factorize **$6x^2 + 8x$

- Identify GCF: $2x$
- Expression becomes: $2x(3x + 4)$

**Factorizing Quadratic Expressions**

Quadratic expressions can often be factorized into two binomials.

**Example: Factorize **$x^2 + 7x + 10$.

- Find two numbers that multiply to $10$ and add to $7$: $2$ and $5$
- Expression factorizes to: $(x + 2)(x + 5)$

**Factorization by Grouping**

When an expression contains four or more terms, grouping them can facilitate factorization.

**Example: Factorize **$x^3 + x^2 - x - 1$**.**

1. Group terms: $(x^3 + x^2) - (x + 1)$

2. Factor out common terms: $x^2(x + 1) - 1(x + 1)$

3. Factor by grouping: $(x + 1)(x^2 - 1)$

4. Notice $(x^2 - 1)$ is a difference of squares: $(x + 1)(x + 1)(x - 1)$

**Difference of Squares**

This technique applies to expressions that can be represented as the difference between two squares.

**Example: Factorize **$a^2 - 25$**.**

- Recognize as a difference of squares: $a^2 - 5^2$
- Factorize: $(a + 5)(a - 5)$

**Practice Questions**

Let's dive into some practice problems with detailed, equation-focused solutions.

**Question 1: Factorize **$3x^2 + 9x$.

- GCF is $3x$.
- Factorize: $3x(x + 3)$

**Question 2: Factorize **$x^2 + 6x + 9$.

- Look for numbers that multiply to $9$ and add to $6$: $3$ and $3$
- Factorize: $(x + 3)^2$

**Question 3: Factorize **$4y^2 - 16$

- Recognize as a difference of squares: $4y^2 - 4^2$
- Factorize: $(2y + 4)(2y - 4)$

**Question 4: Factorize **$x^4 - 16$

- Recognize as a difference of squares: $(x^2)^2 - 4^2$
- Apply difference of squares: $(x^2 + 4)(x^2 - 4)$
- Notice $x^2 - 4$ is also a difference of squares: $(x^2 + 4)(x + 2)(x - 2)$