Expanding & Factorising

• The Distributive Law states that $a(b + c) = ab + ac$. This principle ensures that the multiplier outside the bracket is applied equally to every component within the sum or difference inside.

• For multiple brackets, the law is extended: every term in the first bracket must be multiplied by every term in the second bracket. This is often visualized using an Area Model, where the total area of a large rectangle is the sum of the areas of smaller sub-rectangles.

• When expanding expressions like $(x + y)^2$, the distributive law reveals that the result is $x^2 + 2xy + y^2$, not simply $x^2 + y^2$. The 'middle term' $2xy$ arises from the cross-multiplication of the two distinct variables.

Expanding Brackets

• Single Brackets: Multiply the external term by each internal term. For example, $3x(2x - 5)$ becomes $6x^2 - 15x$.
• Double Brackets: Use a systematic approach like the grid method or FOIL (First, Outer, Inner, Last) to ensure all four pairs of terms are multiplied. Always collect like terms afterward to simplify the expression.
• Triple Brackets: Expand the first two brackets first, simplify the resulting quadratic, and then multiply every term of that quadratic by every term in the third bracket.

Factorising Techniques

• Highest Common Factor (HCF): Identify the largest number and the highest power of each variable that divides into every term. Place this HCF outside the bracket and divide each original term by it to find the internal terms.
• Factorising by Grouping: Used for expressions with four terms. Group the terms into two pairs, factorise the HCF out of each pair, and then factorise out the resulting common bracket.
• Difference of Two Squares (DOTS): Recognize the pattern $a^2 - b^2$. This always factorises into $(a - b)(a + b)$. This is a critical shortcut for simplifying complex algebraic terms.

• Perfect Square vs. DOTS: A perfect square expansion $(a+b)^2$ results in three terms ($a^2 + 2ab + b^2$), whereas the Difference of Two Squares $(a-b)(a+b)$ results in only two terms ($a^2 - b^2$) because the middle terms cancel out.

• The 'Reverse Check': Always verify a factorisation by mentally expanding the brackets. If the expansion does not return the exact original expression, a sign error or an incorrect HCF has likely occurred.

• Factorise Fully: If a question asks to factorise 'fully', check if the terms inside your resulting bracket still share a common factor or if they form a Difference of Two Squares that can be broken down further.

• Sign Awareness: Be extremely careful with negative signs outside brackets. When expanding $-3(x - 4)$, the result is $-3x + 12$. Forgetting to change the sign of the second term is one of the most common ways to lose marks.

• Equating Coefficients: In identity problems, expand the more complex side fully and group terms by their power of $x$. Set the coefficients of $x^2$, $x$, and the constant terms equal to each other to solve for unknowns.

• The Square Trap: Many students incorrectly believe $(x + y)^2 = x^2 + y^2$. Always write out the brackets as $(x + y)(x + y)$ to ensure the middle $2xy$ term is included.

• Incomplete HCF: Students often take out a numerical factor but forget a variable factor (or vice versa). For example, factorising $6x^2 + 12x$ as $6(x^2 + 2x)$ is not 'full' factorisation; the correct HCF is $6x$.

• DOTS Misidentification: The Difference of Two Squares only applies to subtraction. An expression like $x^2 + 9$ cannot be factorised using the DOTS method because it is a sum, not a difference.