Mathematics · Lesson 8

Algebraic processes: factorisation, change of subject, algebraic fractions, polynomials

JAMBWAECNECO

What the syllabus asks

JAMB: polynomials (change of subject of formula, factor and remainder theorems, factorisation, multiplication and division of polynomials), algebraic fractions

WAEC: algebraic processes: expansion and factorisation (common factors, grouping, quadratic expressions, difference of two squares, perfect squares); change of subject of formulae; algebraic fractions (simplification, operations, values for which a fraction is undefined); substitution; polynomials: remainder and factor theorems, factorisation of cubics (NECO/Further Maths overlap)

NECO: algebraic processes: expansion and factorisation (common factors, grouping, quadratic expressions, difference of two squares, perfect squares); change of subject of formulae; algebraic fractions (simplification, operations, values for which a fraction is undefined); substitution; polynomials: remainder and factor theorems, factorisation of cubics (NECO/Further Maths overlap)

What you will learn

Expand products of binomials and simple polynomials accurately.
Factorise by common factor, grouping, difference of squares, perfect squares and trinomials.
Change the subject of formulae, including those with squares, roots and the subject appearing twice.
Simplify, add, subtract, multiply and divide algebraic fractions.
State values for which an algebraic fraction is undefined.
Use the remainder and factor theorems and divide polynomials.
Factorise cubic polynomials completely.

Key points

Expand carefully; (a ± b)² = a² ± 2ab + b²; (a + b)(a − b) = a² − b².
Take out the HCF first, then look for other patterns.
Group four-term expressions in pairs and take out the common bracket.
Difference of squares and perfect squares; a sum of squares does not factorise.
Trinomials: product ac and sum b; split the middle term; group.
Change of subject: undo operations in reverse; collect and factorise when the subject appears twice.
Simplify fractions by cancelling factors, never terms; undefined where the denominator = 0.
Add/subtract with an LCM and brackets; divide by multiplying by the reciprocal.
Remainder theorem: remainder on division by (x − a) is f(a); factor theorem: f(a) = 0 ⇒ (x − a) is a factor.
Factorise cubics by finding one root, dividing, then factorising the quadratic.

Start here: Expanding brackets

Factorising is expanding in reverse, so we begin by expanding confidently.

Single bracket

Multiply every term inside by the term outside: 3x(2x − 5) = 6x² − 15x; −2(4 − y) = −8 + 2y. Watch the sign: a negative outside changes every sign inside.

Two brackets

Each term in the first bracket multiplies each term in the second (often remembered as FOIL: First, Outer, Inner, Last):

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.

Worked example 8.1

Expand and simplify: (a) (2x − 3)(x + 4), (b) (3a − 2b)², (c) (x + 1)(x² − x + 1).

(a) 2x² + 8x − 3x − 12 = 2x² + 5x − 12.
(b) (3a − 2b)(3a − 2b) = 9a² − 6ab − 6ab + 4b² = 9a² − 12ab + 4b². (Not 9a² + 4b²: the middle term is easily forgotten.)
(c) x³ − x² + x + x² − x + 1 = x³ + 1.

2x² + 5x − 12; 9a² − 12ab + 4b²; x³ + 1

Special products to recognise

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²

Worked example 8.2 (mental arithmetic)

Use special products to evaluate (a) 103², (b) 47 × 53.

(a) (100 + 3)² = 10,000 + 600 + 9 = 10,609.
(b) (50 − 3)(50 + 3) = 2500 − 9 = 2491.

10,609; 2491

⚠️ Common mistake: (x + 5)² = x² + 25 ✗. Correct: x² + 10x + 25.

Try it yourself. Expand (3x + 2)(2x − 5).

Show solution

6x² − 15x + 4x − 10 = 6x² − 11x − 10.

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Practice questions

1. Expand (2x − 1)(3x + 4).

Show worked answer

6x² + 5x − 4.

2. Factorise 4x² − 12x + 9.

Show worked answer

(2x − 3)².

3. Factorise 5x² − 13x − 6.

Show worked answer

ac = −30: −15 and 2 → (5x + 2)(x − 3).

4. Make h the subject of V = πr²h/3.

Show worked answer

h = 3V/(πr²).

5. Make x the subject of y = (2x − 1)/(x + 4).

Show worked answer

x = (4y + 1)/(2 − y).

Key words

Expression: A combination of numbers, letters and operations without an equals sign.
Expand: Remove brackets by multiplying out.
Factorise: Write an expression as a product of factors.
HCF: Highest common factor of all the terms.
Difference of two squares: a² − b² = (a − b)(a + b).
Perfect square: An expression that is the square of another, e.g. (x + 3)².
Quadratic expression: An expression of the form ax² + bx + c, a ≠ 0.
Subject of a formula: The letter on its own on one side of the formula.
Algebraic fraction: A fraction whose numerator or denominator contains letters.
LCM: Lowest common multiple, used for common denominators.
Polynomial: A sum of terms in non-negative whole-number powers of x.
Degree: The highest power of x in a polynomial.

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