What is covered in Grade 9 Mathematics Algebraic Expressions — Expand & Factorise?

Multiply polynomials, divide by integers/monomials, product of two binomials, square of binomial, factorisation including difference of squares and trinomials. Section 1.3 covers algebraic fractions (extension content beyond core CAPS).

Is Grade 9 Algebraic Expressions — Expand & Factorise tested in Paper 1 or Paper 2?

Grade 9 Mathematics Algebraic Expressions — Expand & Factorise is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Algebraic Expressions — Expand & Factorise taught in Grade 9 Mathematics?

Algebraic Expressions — Expand & Factorise is taught in Term 2 of Grade 9 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 9 Mathematics Algebraic Expressions — Expand & Factorise on the CAPS curriculum?

Yes. Algebraic Expressions — Expand & Factorise is part of the official CAPS Mathematics curriculum for Grade 9, Term 2 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 9 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 9 Mathematics

Grade 9 · Term 2Mathematics

Algebraic Expressions — Expand & Factorise

We multiply polynomials using the distributive law, including the product of two binomials and the square of a binomial. We then reverse the process through factorisation: finding the HCF, grouping, difference of squares, and trinomial factorisation.

1.1 Products of Polynomials 1.2 Factorisation 1.3 Algebraic Fractions ★ Extension

Weeks 1–2

1.1 Products of Polynomials

Multiply integers and monomials by polynomials
Determine the product of two binomials
Square a binomial: (a ± b)²
Recognise the pattern for difference of squares: (a + b)(a − b) = a² − b²

🌍

Real-World Connection

Expanding brackets is like calculating the area of a tiled floor with two different sections side by side. If one section is (x + 3) metres wide and both sections are (x + 2) metres long, the total area = (x + 2)(x + 3) = x² + 5x + 6 m². Every term in the first bracket multiplies every term in the second — just like every row of tiles covers every column.

Property / Rule

Distributive Law

To multiply a monomial by a polynomial, multiply the monomial by EACH term inside the brackets.

a(b + c + d) = ab + ac + ad

FOIL method for binomial products: First, Outer, Inner, Last.

Product of Two Binomials (FOIL)

(a + b)(c + d) = ac + ad + bc + bd

Multiply every term in the first bracket by every term in the second

Square of a Binomial (Sum)

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

a = first term, b = second term; the middle term 2ab is crucial and often forgotten

Square of a Binomial (Difference)

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

Note: the middle term is NEGATIVE (−2ab), but the result is never negative

Difference of Squares Pattern

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

The middle terms cancel: +ab and −ab. This pattern works in reverse for factorisation.

🚨 Common Mistake

The most common error: (a + b)² ≠ a² + b². There is ALWAYS a middle term: 2ab. Check: (3 + 4)² = 7² = 49, but 3² + 4² = 9 + 16 = 25 ≠ 49.

Worked Examples

Worked Example

Expanding products of binomials

Problem

Expand and simplify: \quad (a)\; (2x + 3)(x - 5) \quad (b)\; (3x - 4)^2

Worked Example

Difference of squares and combined expansion

Problem

Expand: \quad (a)\; (5x + 2y)(5x - 2y) \quad (b)\; (x + 3)^2 - (x - 3)^2

Worked Example

Binomial times trinomial

Problem

(2x - 1)(3x^2 + x - 4)

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

Expand:

3x(2x - 5)

L1 · Knowledge2 marks

Write down the expansion of

(a+b)(a-b)

and identify the pattern.

L2 · Routine Procedures4 marks

Expand and simplify:

(x + 4)(x - 3) - (x - 1)^2

L2 · Routine Procedures3 marks

Expand:

(2x - 3)(2x + 3)

. What pattern do you notice?

L3 · Complex Procedures4 marks

Show that

(x+2)^2 - (x-2)^2 = 8x

without a calculator. Explain WHY the result has no

x^2

term.

L3 · Complex Procedures5 marks

A square room has side length

(x + 5)

metres. A square rug has side length

(x - 1)

metres. Write a simplified expression for the uncovered floor area.

L4 · Problem Solving5 marks

Without expanding, explain why

(99)^2 = (100-1)^2 = 9\,801

. Then use the same method to find

101^2

L4 · Problem Solving4 marks

x + y = 7

and

x - y = 3

, find

x^2 - y^2

WITHOUT finding

x

and

y

separately.

Weeks 2–3

1.2 Factorisation

Factorise algebraic expressions: highest common factor (HCF)
Factorise by grouping in pairs
Factorise difference of two squares
Factorise trinomials of the form ax² + bx + c

🌍

Real-World Connection

Factorisation is the reverse of expansion — it's like going from 'answer' back to 'question'. A contractor who builds rooms of area x² + 7x + 12 m² wants to know the room dimensions. Factorising gives (x + 3)(x + 4) — the length and width. In security, prime factorisation of huge numbers protects your bank card PIN from hackers.

Definition

Highest Common Factor (HCF)

The largest expression that divides exactly into all terms. Always look for the HCF first — it simplifies everything that follows.

\text{e.g. } 6x^2 + 9x = 3x(2x + 3) \quad \text{HCF} = 3x

Difference of Squares: a² − b² splits into two conjugate binomials (a + b)(a − b).

Difference of Squares

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

Both terms must be perfect squares and subtracted (not added)

For trinomials x² + bx + c: find two integers p and q such that p × q = c AND p + q = b. Then x² + bx + c = (x + p)(x + q).

💡 Tip

Factorisation checklist: (1) Take out the HCF. (2) Look for Difference of Squares (2 terms). (3) Try grouping (4 terms). (4) Try trinomial factorisation (3 terms). Always verify by expanding your answer.

Worked Examples

Worked Example

Factorising using multiple strategies

Problem

Factorise completely: \quad (a)\; 12x^3 - 8x^2 + 4x \quad (b)\; x^2 - 49 \quad (c)\; x^2 + 5x + 6

Worked Example

Factorising trinomials ax² + bx + c (a ≠ 1)

Problem

Factorise: \quad (a)\; 6x^2 + 11x + 4 \quad (b)\; 2x^2 - 7x + 3

Worked Example

Factorising by grouping and mixed types

Problem

Factorise completely: \quad (a)\; 3ax - 6ay + bx - 2by \quad (b)\; 5x^2 - 20

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

Factorise by taking out the HCF:

15x^2 - 10x

L1 · Knowledge2 marks

Factorise:

x^2 - 64

L2 · Routine Procedures3 marks

Factorise completely:

x^2 - 3x - 18

L2 · Routine Procedures3 marks

Factorise by grouping:

3x^2 + 6x + x + 2

L3 · Complex Procedures4 marks

Factorise completely:

2x^3 - 8x

L3 · Complex Procedures4 marks

The area of a rectangle is

x^2 + 2x - 15

. Find possible expressions for the length and width.

L4 · Problem Solving5 marks

Factorise:

4x^2 - 12x + 9 - 25y^2

L4 · Problem Solving5 marks

Simplify:

\dfrac{x^2 - 9}{x^2 + x - 12}

Week 3

1.3 Algebraic Fractions ★ Extension

Simplify algebraic fractions by factorising the numerator and denominator
Multiply and divide algebraic fractions
Add and subtract algebraic fractions with monomial and binomial denominators

🌍

Real-World Connection

Algebraic fractions work exactly like number fractions — you can only cancel common FACTORS, not common terms. Just as 8/12 = 2/3 (cancel the factor 4), so (x²−4)/(x+2) = x−2 (cancel the factor (x+2)). The golden rule: factorise first, then cancel.

ℹ️ Note

EXTENSION: This section goes beyond the core CAPS Grade 9 syllabus. Mastering algebraic fractions now provides a significant advantage in Grade 10 algebra and prepares you for rational expressions in higher mathematics.

To simplify an algebraic fraction: (1) Factorise the numerator completely. (2) Factorise the denominator completely. (3) Cancel any common FACTORS (not terms). (4) State any values of the variable that make the denominator zero.

⚠️ Warning

You may only cancel FACTORS — entire expressions that multiply the whole numerator or denominator. NEVER cancel individual terms: (x+3)/3 ≠ x. The 3 is a TERM in the numerator, not a factor.

Multiplying Algebraic Fractions

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Factorise all expressions first, then cancel before multiplying

Adding Fractions — Same Denominator

\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

Add numerators; keep the common denominator

Adding Fractions — Different Denominators

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Find the LCD, convert, then add numerators

Worked Examples

Worked Example

Simplifying and adding algebraic fractions

Problem

Simplify: \; \dfrac{x^2-4}{x+2} \qquad \text{Then add: } \dfrac{3}{x} + \dfrac{2}{x+1}

Worked Example

Simplifying algebraic fractions by factorising

Problem

Simplify: \quad \dfrac{x^2-9}{x^2+x-6}

Worked Example

Multiplying and dividing algebraic fractions

Problem

Simplify: \quad \dfrac{2x^2-8}{x^2+2x} \div \dfrac{x-2}{3x}

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

Simplify:

\dfrac{6x^2}{3x}

L1 · Knowledge2 marks

Simplify:

\dfrac{x^2 - 25}{x - 5}

, stating any restrictions.

L2 · Routine Procedures4 marks

Simplify:

\dfrac{x^2 + 3x}{x^2 - x - 12}

L2 · Routine Procedures4 marks

Calculate:

\dfrac{2}{x+1} - \dfrac{1}{x-1}

L3 · Complex Procedures4 marks

Simplify:

\dfrac{x^2-1}{x^2+3x+2}

L3 · Complex Procedures5 marks

Simplify:

\dfrac{x}{x-2} + \dfrac{4}{x^2-4}

L4 · Problem Solving5 marks

Simplify:

\dfrac{x^2-x-6}{x^2-4} \div \dfrac{x^2-9}{x+2}

L4 · Problem Solving4 marks

Show that

\dfrac{3}{x-2} - \dfrac{3}{x+2} = \dfrac{k}{x^2-4}

and find the constant value of

k

Algebraic Equations