Algebraic Expressions & Factoring

Factoring is like reverse multiplication -- instead of expanding brackets, you put them back. This skill is essential for solving equations, simplifying fractions, and working with higher-level algebra in Grade 11.

What is Factorization?

Factorization means writing an expression as a product of its factors. For example, 12 = 2 x 2 x 3. Similarly, x^2 + 5x + 6 = (x + 2)(x + 3).

Key Factoring Techniques

1. Common Factor (HCF Method)

Look for the highest common factor of all terms and take it out.

Example: 6x^2 + 9x = 3x(2x + 3)

2. Grouping

When there are four terms, group them in pairs and factor each pair.

Example: ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)

3. Difference of Squares

Formula: a^2 - b^2 = (a + b)(a - b)

Example: x^2 - 16 = (x + 4)(x - 4)

4. Perfect Square Trinomials

Formulas: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2

Example: x^2 + 6x + 9 = (x + 3)^2

5. Trinomial Factoring (Splitting the Middle Term)

For ax^2 + bx + c, find two numbers that multiply to give ac and add to give b.

Example: x^2 + 5x + 6

We need two numbers that multiply to 6 and add to 5. Those are 2 and 3.

x^2 + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 2)(x + 3)

HCF and LCM of Algebraic Expressions

Just like numbers, algebraic expressions have HCF and LCM.

Example: Find HCF and LCM of 4x^2y and 6xy^2.

• HCF: Take lowest powers of common factors = 2xy
• LCM: Take highest powers of all factors = 12x^2y^2

Useful Check: HCF x LCM = Product of the two expressions.

Did You Know? Factoring is used in cryptography -- the security of online banking and digital payment apps relies on the difficulty of factoring extremely large numbers.

Key Takeaways

• Always check for a common factor first
• Recognize patterns: difference of squares, perfect squares, trinomials
• For trinomials, split the middle term by finding the right pair of numbers
• HCF uses lowest powers; LCM uses highest powers