U13 Identities and Factorization: Essential Formulas

This section covers the fundamental identities used for expansion and factorization, which are crucial for simplifying algebraic expressions and solving equations in the DSE examination.

1 Basic Algebraic Identities

Perfect Square Identities

These identities are used to expand squares of binomials and are frequently applied in reverse for factorization.

$$ (a + b)^2 = a^2 + 2ab + b^2 $$

$$ (a - b)^2 = a^2 - 2ab + b^2 $$

Difference of Two Squares

A powerful identity for factorization. It states that the difference between two squares is equal to the product of the sum and difference of the two terms.

$$ a^2 - b^2 = (a + b)(a - b) $$

2 Factorization Methods

Taking Out Common Factors

The most basic method. Identify the highest common factor (HCF) of all terms and factor it out.

For example, in the expression $ax + ay$, the common factor is $a$, so it factors to $a(x + y)$.

Factorization by Grouping Terms

Used for expressions with four or more terms. Group terms with common factors, then factor each group.

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

Factorization Using Identities

Recognize the structure of an expression that matches a known identity (like the ones above) and apply it in reverse.

For instance, $x^2 - 9y^2$ is a difference of two squares $(x)^2 - (3y)^2$, so it factors to $(x + 3y)(x - 3y)$.

3 Further Identities

Sum and Difference of Two Cubes

These identities are useful for factorizing cubic expressions.

$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$

$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$

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