Algebraic Expressions

Master the language of algebra through variables, terms, and operations

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and provides foundational skills for advanced algebra. Each section includes interactive games and quizzes to test your understanding.

Learning Outcomes

  • Identify variables, constants, terms, and coefficients
  • Simplify expressions by combining like terms
  • Expand algebraic expressions using distributive property
  • Factorise expressions using various methods
  • Work with algebraic fractions
  • Apply algebraic skills to solve problems
  • Understand and use binomial products

1. Basic Algebraic Concepts

V

Variables

  • Symbols representing unknown values
  • Usually letters: x, y, z, a, b
  • Can change value
  • Example: x in 3x + 2
C

Constants

  • Fixed numerical values
  • Don't change
  • Example: 2 in 3x + 2
T

Terms

  • Parts separated by + or -
  • Example: 3x, -2y, +5
  • Like terms: Same variables & powers

Simplifying Expressions

Example

Simplify: 3x + 2y - x + 5y

Solution
  • Identify like terms: x terms: 3x, -x | y terms: 2y, 5y
  • Combine x terms: 3x - x = 2x
  • Combine y terms: 2y + 5y = 7y
  • Final answer: 2x + 7y

Quiz 1 - Basic Concepts

What do we call a symbol that represents an unknown value?

A) Constant
B) Variable
C) Coefficient
D) Term

2. Expanding Algebraic Expressions

Distributive Property

a(b + c) = ab + ac

Multiply each term inside by term outside

FOIL Method

For (x + a)(x + b):

  • First: x × x = x²
  • Outer: x × b = bx
  • Inner: a × x = ax
  • Last: a × b = ab
= x² + (a+b)x + ab

Expanding Examples

Example 1

Expand: 2x(x - 3)

Solution
2x × x - 2x × 3 = 2x² - 6x
Example 2

Expand: (x + 2)(x - 5)

Solution
x² - 5x + 2x - 10 = x² - 3x - 10

Special Products

  • (x + a)² = x² + 2ax + a²
  • (x - a)² = x² - 2ax + a²
  • (x + a)(x - a) = x² - a²

Quiz 2 - Expanding

Expand: (x + 3)(x - 2)

A) x² + 5x - 6
B) x² + x - 6
C) x² - 5x + 6
D) x² - x + 6

3. Factorisation Methods

HCF (Highest Common Factor)

Find common factor in all terms

Factorise: 6x² + 9x

HCF = 3x
= 3x(2x + 3)

Difference of Squares

a² - b² = (a + b)(a - b)

Factorise: x² - 16

= x² - 4² = (x + 4)(x - 4)

Trinomial Factorisation

Example

Factorise: x² + 5x + 6

Solution
  • Need p and q where: p + q = 5, p × q = 6
  • Factors of 6: 2 and 3 (2 + 3 = 5)
  • Factorised form: (x + 2)(x + 3)

Quiz 3 - Factorisation

Factorise: x² - 9

A) (x - 3)²
B) (x + 3)²
C) (x - 3)(x + 3)
D) (x - 9)(x + 1)

4. Algebraic Fractions

Simplifying Fractions

Factorise numerator & denominator, cancel common factors

Simplify: (x² - 4)/(x + 2)

x² - 4 = (x+2)(x-2)
= [(x+2)(x-2)]/(x+2) = x - 2

Addition/Subtraction

Same denominator: add/subtract numerators

x/(x+1) + 2/(x+1)

= (x + 2)/(x + 1)

Fractions with Different Denominators

Example

Simplify: x/(x-2) - 3/(x+1)

Solution
  • LCD = (x-2)(x+1)
  • = [x(x+1) - 3(x-2)]/[(x-2)(x+1)]
  • = [x² + x - 3x + 6]/[(x-2)(x+1)]
  • = (x² - 2x + 6)/[(x-2)(x+1)]

Quiz 4 - Fractions

Simplify: (x² - 9)/(x - 3)

A) x - 3
B) x + 3
C) x
D) 3

Practice & Assess

Match - Terms to Definitions

Variable
symbol for unknown
Constant
fixed value
Term
part separated by +/-
Coefficient
number multiplying variable

Fill - Special Product

(x + 3)² = x² + 6x + ___

5. Common Algebraic Errors

Error 1

Distributive errors: 2(x+3) ≠ 2x+3

Correct: 2(x+3) = 2x + 6

Error 2

Squaring errors: (x+3)² ≠ x² + 9

Correct: (x+3)² = x² + 6x + 9

6. Practice Problems

Question 1

Simplify: 5a² - 3a + 2a² + 4a - 1

Question 2

Expand: (3x - 4)(2x + 5)

Question 3

Factorise: 9y² - 25

Answers

Q1: 7a² + a - 1 | Q2: 6x² + 7x - 20 | Q3: (3y + 5)(3y - 5)

Key Terms

Variable Constant Term Coefficient Like terms Distributive FOIL Binomial HCF Difference of squares Trinomial LCD
Exponents and Surds Equations and Inequalities