7 min readConcept sheetShort answer or part of a structured question

Algebraic expressions and factoring

Algebra uses letters for unknown amounts. Expand to remove brackets; factorise to put them back. Always check factors by expanding.

Terme.g. 5x² (number × letters)

Like termsSame letters, same powers

Key identitya² − b² = (a + b)(a − b)

Self-checkExpand your factors

Key identities

(a + b)²a² + 2ab + b²

(a − b)²a² − 2ab + b²

a² − b² (difference of two squares)(a + b)(a − b)

(x + p)(x + q)x² + (p + q)x + pq

a(b + c)ab + ac

Order of working (BODMAS)Brackets, Orders (powers), Division and Multiplication, Addition and Subtraction

Factorising: which method?

Example of the three-term rule: 2x² + 7x + 3 has ac = 6 and b = 7, so use 6 and 1.

Simplify and expand

Problem

(a) Simplify 3(2x − 5) − 2(x − 4). (b) Expand (2x + 3)(x − 4).

Working

(a) Open brackets: 6x − 15 − 2x + 8. (A minus times a minus gives plus: −2 × −4 = +8.)
Collect like terms: 6x − 2x = 4x and −15 + 8 = −7.
(b) Multiply each term in the first bracket by each term in the second.
2x × x = 2x²; 2x × (−4) = −8x; 3 × x = 3x; 3 × (−4) = −12.
Collect: 2x² − 8x + 3x − 12 = 2x² − 5x − 12.
Check (b) with x = 1: (5)(−3) = −15 and 2 − 5 − 12 = −15. ✓

Answer

(a) 4x − 7. (b) 2x² − 5x − 12.

Factorise fully

Problem

Factorise (a) 6x² − 15x, (b) x² − 7x + 12, (c) 4x² − 25, (d) 2x² + 7x + 3.

Working

(a) Common factor 3x: 6x² − 15x = 3x(2x − 5).
(b) Need two numbers with product +12 and sum −7: −3 and −4. So (x − 3)(x − 4).
(c) Difference of two squares: (2x)² − 5² = (2x + 5)(2x − 5).
(d) ac = 2 × 3 = 6; two numbers with product 6 and sum 7 are 6 and 1.
Split the middle: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Check (d): (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. ✓

Answer

(a) 3x(2x − 5) (b) (x − 3)(x − 4) (c) (2x + 5)(2x − 5) (d) (2x + 1)(x + 3)

An expression in business

Problem

A Galle soap maker sells x bars at Rs. 50 each. Costs are Rs. 3,000 fixed plus Rs. 20 a bar. Write profit as an expression, factorise it, and find when profit is zero.

Working

Revenue R = 50x. Cost C = 3,000 + 20x.
Profit P = R − C = 50x − (3,000 + 20x) = 50x − 3,000 − 20x = 30x − 3,000.
Common factor 30: P = 30(x − 100).
P = 0 when x − 100 = 0, so x = 100.
Check: 50 × 100 = 5,000 and 3,000 + 20 × 100 = 5,000. ✓

Answer

P = 30x − 3,000 = 30(x − 100); profit is zero at 100 bars.

In Sri Lankan business

A Colombo three-wheeler meter works like the expression F = a + bk: a starting charge a plus a rate b for each kilometre k. Payroll, pricing and cost sheets in every Sri Lankan firm are built from such expressions.

Exam angle

Short answer: “Factorise x² − 9x + 20.” Two numbers with product 20 and sum −9 are −4 and −5: (x − 4)(x − 5).

Short answer: “Expand (3x − 2)².” 9x² − 12x + 4 (use (a − b)² = a² − 2ab + b²).

Structured question: “Factorise fully: (a) 3x² − 12, (b) 6x² − x − 2.” (a) 3(x² − 4) = 3(x + 2)(x − 2). (b) ac = −12, numbers −4 and 3: (2x + 1)(3x − 2).

Remember it

Factorising order: Common factor, Difference of squares, Trinomial, Grouping: "Can Dad Take Grapes?" Then expand to check.