Long before modern symbols, people solved problems that required multiplying lengths and collecting areas — the same ideas behind expanding and factorising. Babylonian scribes (around 1800 BCE) recorded rectangle problems on clay tablets: given area and side relationships, find unknown lengths. Their recipes are early cousins of “expand, then simplify.”
In the 9th century, al-Khwarizmi organised methods for equations and products in Arabic texts that later spread to Europe. Renaissance mathematicians such as Viète introduced letters for unknowns and constants, making it easier to see patterns like \(a^2 - b^2\). By the 17th–18th centuries, polynomial algebra was central to calculus and science; factorisation became a routine step before solving equations or simplifying formulas.
Draw a rectangle with width \(a+b\) and height \(c+d\). The total area is \((a+b)(c+d)\). Split the rectangle into four smaller rectangles with areas \(ac\), \(ad\), \(bc\), and \(bd\). The sum of partial areas equals the whole: \[ (a+b)(c+d) = ac + ad + bc + bd. \] This is the distributive law in two dimensions — the same picture as in the area-model widget above.
Set \(c = a\) and \(d = b\) in the area formula for a square of side \(a+b\): \[ (a+b)^2 = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2. \] For a square of side \(a-b\) (with \(a > b\)), remove a strip of width \(b\) from a square of side \(a\); the remaining L-shape rearranges to show \[ (a-b)^2 = a^2 - 2ab + b^2. \] Difference of squares: a large square of side \(a\) minus a small square of side \(b\) leaves two congruent rectangles, which rearrange to a rectangle of dimensions \((a+b)\) by \((a-b)\): \[ a^2 - b^2 = (a+b)(a-b). \]
Factorise \(3x^2 + 12x - 2x - 8\).
Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
Checkpoint
In \(4x^3 - 2x^2 + 5\), what is the degree? Which terms are like terms with \(4x^3\)?
Checkpoint
Expand \((2x-1)(x+4)\) without skipping steps. How many terms do you get before collecting?
Checkpoint
How would you factorise \(x^2 - 16\)? Which identity applies?
Checkpoint
Expand \(x(x+1) - (x+1)\). What common factor appears? What is the factorised form?
A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Its area is \[ (x+3)(2x-1) = 2x^2 + 5x - 3 \quad\text{cm}^2. \] The perimeter is \(2\bigl((x+3)+(2x-1)\bigr) = 2(3x+2) = 6x+4\) cm. Expansion links algebraic descriptions of sides to measurable quantities.
If a shop sells \(n\) items at price \((50 - 0.5n)\) dollars each, revenue is \(n(50 - 0.5n) = 50n - 0.5n^2\). Expanding a product of two linear factors often appears before finding maximum revenue (a quadratic in \(n\)).
Notice that \(11^2 = 121\), \(111^2 = 12321\). In general, expanding \((10^k + 10^{k-1} + \cdots + 1)^2\) explains palindromic squares — a playful link between identities and arithmetic patterns.
Work each question, then open the solution to check your reasoning. Difficulty increases through the sets.
Answer: \(3x + 6\).
Steps: \(3 \cdot x + 3 \cdot 2 = 3x + 6\).
Answer: \(7x - 1\).
Explanation: Combine like terms \(2x + 5x = 7x\).
Answer: \(3(2x + 3)\).
Steps: HCF is \(3\); \(6x \div 3 = 2x\), \(9 \div 3 = 3\).
Answer: \(x^2 + 2x - 8\).
Steps: \(x^2 - 2x + 4x - 8 = x^2 + 2x - 8\).
Answer: \((x+3)(x-3)\).
Explanation: Difference of squares: \(x^2 - 3^2\).
Answer: \((x+3)^2\).
Explanation: Perfect square: \(3 \times 3 = 9\), \(3 + 3 = 6\).
Answer: \((2x-1)(x+3)\).
Steps: \(2x(x+3) - 1(x+3) = (2x-1)(x+3)\).
Answer: \((2x+1)(x+3)\).
Steps: \(ac = 6\); split \(7x = 6x + x\); \(2x(x+3) + 1(x+3) = (2x+1)(x+3)\).
Answer: \((x+2)(x-1)\) after factorising (expansion gives \(x^2 + x - 2\)).
Steps: Factor common \((x+2)\): \((x+2)(x-1)\). Check: \((x+2)(x-1) = x^2 + x - 2\).
Answer: \((2x^2 + 5x - 3)\) cm².
Calculation: \((x+3)(2x-1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3\).