# Factorization of Algebraic Expressions - Methods and Examples

## What Is Factorization of Algebraic Expressions?

Factorization of an algebraic expression is the process of writing the expression as a **product of two or more factors**, where multiplying those factors back together returns the original expression. A **factor** is a quantity that divides another exactly, with no remainder.

## How Do You Factorize an Algebraic Expression?

The four methods that cover almost everything at this level:

- **Common factor method**
- **Grouping**
- **Algebraic identities**
- **Splitting the middle term**

### Method 1: The common factor

Find the largest factor every term shares.

### Method 2: Grouping

When an expression has four terms and no factor is common to all four, group them in pairs, factor each pair, then factor out the bracket they now share.

### Method 3: Algebraic identities

Some expressions match a standard algebraic identity.

| Identity | Factored form |
| --- | --- |
| a² + 2ab + b² | (a + b)² |
| a² - 2ab + b² | (a - b)² |
| a² - b² | (a + b)(a - b) |
| a³ + b³ | (a + b)(a² - ab + b²) |
| a³ - b³ | (a - b)(a² + ab + b²) |

### Method 4: Splitting the middle term

For a quadratic ax² + bx + c, find two numbers that multiply to ac and add to b.

## Examples of Factorization of Algebraic Expressions

### Example 1: Factorize 5z³ - 10z².
**Final answer:** 5z²(z - 2).

### Example 2: Factorize x² - 10x + 25.
**Final answer:** (x - 5)².

### Example 3: Factorize x² - 4.
**Final answer:** (x + 2)(x - 2).

### Example 4: Factorize x³ + 3x² + 2x + 6 by grouping.
**Final answer:** (x + 3)(x² + 2).

### Example 5: Factorize 2x² + 7x + 3 by splitting the middle term.
**Final answer:** (x + 3)(2x + 1).

### Example 6: Factorize a³ - 8.
**Final answer:** (a - 2)(a² + 2a + 4).

## Where Factorization Earns Its Keep

- **Solving equations.**
- **Simplifying fractions.**
- **Revealing structure.**

## Where Solutions Go Off the Rails

### Mistake 1: Not pulling out the full common factor.
### Mistake 2: Forgetting to factor completely.
### Mistake 3: Sign errors when splitting the middle term.

## Make the Multiply-Back Check a Habit

Teaching factorization with the multiply-back check helps students catch mistakes themselves before submitting.

## Conclusion

- **Factorization of algebraic expressions** rewrites an expression as a product of its factors.
- The four core methods are the common factor, grouping, algebraic identities, and splitting the middle term.
- The most common mistakes are leaving a shared variable inside, not factoring completely, and mismatching signs when splitting the middle term.
- Factoring is the gateway to solving equations.