What is covered in Grade 10 Mathematics Algebraic Expressions?

Real numbers (rational/irrational, surds), products (binomial × trinomial), factorisation (including sum/difference of cubes), and simplification of algebraic fractions with cube denominators.

Is Grade 10 Algebraic Expressions tested in Paper 1 or Paper 2?

Grade 10 Mathematics Algebraic Expressions is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Algebraic Expressions taught in Grade 10 Mathematics?

Algebraic Expressions is taught in Term 1 of Grade 10 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 10 Mathematics Algebraic Expressions on the CAPS curriculum?

Yes. Algebraic Expressions is part of the official CAPS Mathematics curriculum for Grade 10, Term 1 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 10 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 10 Mathematics

Grade 10 · Term 1Mathematics

Algebraic Expressions

We open Grade 10 Mathematics by mapping the real number system, then build every algebraic skill needed for the year: multiplying polynomials, factorising completely (including cubes), and simplifying algebraic fractions with any type of denominator.

1.1 Real Numbers — Rational, Irrational & Surds 1.2 Products — Polynomials & Special Products 1.3 Factorisation 1.4 Algebraic Fractions

Week 1

1.1 Real Numbers — Rational, Irrational & Surds

Understand that real numbers can be rational or irrational
Establish between which two integers a given simple surd lies
Round real numbers to an appropriate degree of accuracy

🌍

Real-World Connection

Every measurement you make — a length, a mass, a temperature — lands somewhere on the real number line. Some land exactly on fractions (rational); others, like the diagonal of a 1 m square ($\sqrt{2}$ m), never terminate or repeat (irrational). Knowing which type you have determines how precisely you can express it.

Definition

Rational number

A number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ . This includes: integers, terminating decimals, and recurring decimals.

\mathbb{Q} = \left\{\frac{a}{b} : a \in \mathbb{Z},\; b \in \mathbb{Z},\; b \neq 0\right\}

Definition

Irrational number

A number that CANNOT be written as a fraction of two integers. Its decimal expansion is non-terminating AND non-recurring. Examples: $\sqrt{2},\; \sqrt{5},\; \pi,\; \sqrt[3]{9}$ .

\sqrt{n} \text{ is irrational when } n \text{ is not a perfect square}

Summary: The Real Number System

Natural numbers $\mathbb{N}$ : $\{1; 2; 3; 4; \ldots\}$
Whole numbers $\mathbb{N}_0$ : $\{0; 1; 2; 3; \ldots\}$
Integers $\mathbb{Z}$ : $\{\ldots; -2; -1; 0; 1; 2; \ldots\}$
Rational numbers $\mathbb{Q}$ : all fractions, terminating and recurring decimals
Irrational numbers: non-terminating, non-recurring decimals (e.g. $\pi$ , $\sqrt{3}$ )
Real numbers $\mathbb{R}$ : all rational AND irrational numbers combined
Non-real: $\sqrt{-4}$ (even root of negative) and division by zero — these have no place on the number line

Property / Rule

Locating a surd between consecutive integers

Find the two perfect squares (or perfect cubes) that bracket the radicand. Take the root of each bracket to find the two integers.

\text{Since } 9 < 12 < 16,\text{ we get } \sqrt{9} < \sqrt{12} < \sqrt{16} \Rightarrow 3 < \sqrt{12} < 4

Property / Rule

Rounding to decimal places

Count to the required decimal place. Look at the NEXT digit: if it is 0–4, drop it (round down); if it is 5–9, add 1 to the last kept digit (round up).

4{,}31\mathbf{4}37 \xrightarrow{2 \text{ d.p.}} 4{,}31 \qquad 1{,}77\mathbf{7}77 \xrightarrow{3 \text{ d.p.}} 1{,}778

🚨 Common Mistake

Do NOT confuse $\sqrt{-9}$ with $-\sqrt{9}$ . The first is non-real (square root of a negative). The second is $-3$ , a perfectly real negative integer.

Worked Examples

Worked Example

Classify real numbers

Problem

\sqrt{16}

Worked Example

Place a surd between consecutive integers

Problem

\sqrt[3]{20}

Worked Example

Show a recurring decimal is rational

Problem

0{,}\overline{27}

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

State whether

\sqrt{25}

is rational or irrational. Give a reason.

L1 · Knowledge2 marks

Which set of numbers does

-7

belong to? Select ALL that apply: natural, whole, integer, rational, real.

L2 · Routine Procedures3 marks

Between which two consecutive integers does

\sqrt{50}

lie? Do NOT use a calculator.

L2 · Routine Procedures3 marks

Round off

3{,}14159265

to (a) 3 decimal places and (b) 5 decimal places.

L3 · Complex Procedures4 marks

Show that

0{,}\overline{142857}

is rational. (Hint: the block has 6 digits.)

L3 · Complex Procedures4 marks

State whether each is rational, irrational, or non-real: (a)

\frac{\pi}{2}

\quad (b)

\sqrt{-1}

\quad (c)

\sqrt[3]{-27}

\quad (d)

\sqrt{0{,}49}

L4 · Problem Solving4 marks

Find a rational number between

\sqrt{3}

and

\sqrt{5}

. Show your reasoning without using a calculator.

L4 · Problem Solving4 marks

A square has area

72

cm². Give the exact length of its side and state whether it is rational or irrational. Between which two consecutive integers does it lie?

Week 2

1.2 Products — Polynomials & Special Products

Multiply a binomial by a trinomial
Identify and apply the special products: $(a+b)^2$, $(a-b)^2$, $(a+b)(a-b)$

🌍

Real-World Connection

A tile contractor quoting for a rectangular room of length $(x+5)$ m and width $(x+3)$ m must price EVERY sub-rectangle: the $x^2$ main area, the two $x$-strips, and the $3\times5$ corner — missing one means the quote is wrong. Polynomial multiplication works exactly the same way: every term in the first factor pairs with every term in the second.

Property / Rule

Distributive Law for Binomial × Trinomial

Multiply every term in the first bracket by every term in the second bracket, then collect like terms.

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

Perfect Square (sum)

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

a, b are any algebraic expressions

Perfect Square (difference)

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

a, b are any algebraic expressions

Difference of Squares

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

a, b are any algebraic expressions

🚨 Common Mistake

A very common error: $(a+b)^2 \neq a^2+b^2$ . The middle term $2ab$ is always present. Write it out in full each time until it is automatic.

Worked Examples

Worked Example

Binomial × Trinomial

Problem

(2x-3)(x^2+5x-4)

Worked Example

Special Products

Problem

(3x+2)^2

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

State the identity for

(a-b)^2

L1 · Knowledge2 marks

Expand

(x+3)(x-3)

L2 · Routine Procedures3 marks

Expand and simplify

(2x+5)(3x-2)

L2 · Routine Procedures4 marks

Expand and simplify

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

L3 · Complex Procedures5 marks

Expand and simplify

(3a-2b)^2 - (3a+2b)(3a-2b)

L3 · Complex Procedures4 marks

Show that

(n+1)^2 - n^2 = 2n+1

for all integers

n

L4 · Problem Solving3 marks

Find

101^2 - 99^2

without a calculator.

L4 · Problem Solving4 marks

(x+k)^2 = x^2+10x+m

, find

k

and

m

Week 3

1.3 Factorisation

Factorise using HCF, difference of squares, trinomials (a=1 and a≠1)
Factorise by grouping in pairs
Factorise a sum of two cubes: $a^3+b^3 = (a+b)(a^2-ab+b^2)$
Factorise a difference of two cubes: $a^3-b^3 = (a-b)(a^2+ab+b^2)$

🌍

Real-World Connection

A locksmith can recreate a key from a lock. Given the 'lock' $x^2 - 5x + 6$, factorisation finds the two key-cuts $(x-2)(x-3)$ — the pair of numbers whose product is $6$ and sum is $-5$. The sum and difference of cubes are master-key patterns that, once memorised, open any cubic lock instantly.

Property / Rule

Highest Common Factor (HCF)

Always look for a common factor first. Include both the highest shared coefficient and the lowest power of each shared variable.

6x^3 - 4x^2 = 2x^2(3x-2)

Property / Rule

Difference of Two Squares

Any expression of the form $a^2-b^2$ factors into two conjugate brackets. Both terms must be perfect squares with a minus sign between them.

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

Property / Rule

Trinomial Factorisation (a=1)

For $x^2+bx+c$ , find two numbers $p$ , $q$ such that $pq=c$ and $p+q=b$ .

x^2+bx+c = (x+p)(x+q)

Property / Rule

Trinomial Factorisation (a≠1)

For $ax^2+bx+c$ , find two numbers whose product is $ac$ and whose sum is $b$ ; split the middle term and factorise by grouping.

6x^2+7x-3 \xrightarrow{ac=-18,\;\text{sum}=7} 6x^2+9x-2x-3

Property / Rule

Factorisation by Grouping in Pairs

When an expression has four terms, arrange them into two pairs and factorise a common factor from each pair. If the same bracket appears in both groups, factor it out.

ax+bx+ay+by = x(a+b)+y(a+b) = (x+y)(a+b)

Sum of two cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

The binomial factor uses +; the trinomial middle term uses −.

Difference of two cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

The binomial factor uses −; the trinomial middle term uses +.

💡 Tip

Memory aid for the cube formulae: SOAP — Same sign as the binomial, Opposite sign in the trinomial middle term, Always Positive last term. Check by expanding back.

Worked Examples

Worked Example

Factorising a Trinomial (a≠1)

Problem

6x^2 + 7x - 3

Worked Example

Sum and Difference of Cubes

Problem

8x^3 + 27

Worked Example

Factorisation by Grouping in Pairs

Problem

x^3+2x^2+3x+6

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

State the formula for the difference of two cubes.

L1 · Knowledge2 marks

Factorise

x^2-25

L2 · Routine Procedures3 marks

Factorise

x^3-x^2+2x-2

by grouping in pairs.

L2 · Routine Procedures3 marks

Factorise

27x^3-1

L3 · Complex Procedures4 marks

Factorise completely:

2x^3+2000y^6

L3 · Complex Procedures4 marks

Factorise completely:

3x^3-48x

L4 · Problem Solving5 marks

Factorise

x^6-y^6

completely in two different ways.

L4 · Problem Solving4 marks

Show that

x^3+1

is divisible by

(x+1)

by factorising, and find the other factor.

Week 5

1.4 Algebraic Fractions

Simplify algebraic fractions by factorising numerator and denominator
Multiply and divide algebraic fractions
Add and subtract algebraic fractions including those with cube denominators

🌍

Real-World Connection

Simplifying an algebraic fraction is exactly like simplifying a numeric fraction: factorise, then cancel factors common to both top and bottom. When denominators involve cubes you need the cube-factorisation skill from the previous section — that is why they were taught first.

Property / Rule

Simplifying Algebraic Fractions

Factorise numerator and denominator completely, then cancel any factors that appear in both. State restrictions where denominator = 0.

\frac{x^2-4}{x^2-x-2} = \frac{(x+2)(x-2)}{(x-1)(x+2)} = \frac{x-2}{x-1},\quad x\neq-2,1

Property / Rule

Multiplying and Dividing

Multiply: factorise all four expressions, cancel across numerators and denominators, then multiply. Divide: multiply by the reciprocal of the second fraction.

\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c}

Property / Rule

Adding and Subtracting — LCD

Factorise all denominators to find the LCD. Convert each fraction to the LCD, then add/subtract numerators. NEVER cancel individual terms from a sum.

\frac{1}{x-2}+\frac{3}{x+1}: \text{LCD}=(x-2)(x+1)

⚠️ Warning

You can ONLY cancel entire factors (brackets). NEVER cancel individual terms: $\dfrac{x+3}{x+5} \neq \dfrac{3}{5}$ . This is one of the most common errors in Grade 10.

Worked Examples

Worked Example

Simplify an Algebraic Fraction

Problem

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

Worked Example

Adding Fractions with a Cube Denominator

Problem

\dfrac{x+2}{4x^2-2x+1} - \dfrac{8x+1}{8x^3+1}

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

For which values of

x

\dfrac{3}{(x-2)(x+1)}

undefined?

L2 · Routine Procedures4 marks

Simplify

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

L2 · Routine Procedures4 marks

Simplify

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

L3 · Complex Procedures4 marks

Simplify

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

L3 · Complex Procedures5 marks

Simplify

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

L4 · Problem Solving6 marks

Simplify

\dfrac{x}{x-2}+\dfrac{4}{x^2-4}-\dfrac{1}{x+2}

L4 · Problem Solving5 marks

Simplify

\dfrac{x+2}{4x^2-2x+1}-\dfrac{1}{2x+1}

given that

8x^3+1=(2x+1)(4x^2-2x+1)

Exponents, Equations & Inequalities