What this chapter is about
This chapter invites you to explore algebra as a playful tool rather than a set of rigid rules. You will work with algebraic expressions, learn how to simplify them, and see how letters (variables) can represent numbers in patterns and puzzles. The focus is on building comfort with expressions, equations, and identities through observation and practice.
By the end of this chapter, you should be able to form algebraic expressions from word problems, add, subtract and multiply expressions confidently, and recognise standard algebraic identities. These skills prepare you for solving harder equations and for topics like factorisation that come later in mathematics.
Algebra connects arithmetic to general reasoning. When you write 2n + 1 for any odd number, you capture every odd number at once. This power of generalisation is what makes algebra so useful in science, finance, and everyday problem-solving.
Key ideas
- A variable (like x, y, n) is a symbol that can take different numerical values; a constant is a fixed number.
- An algebraic expression is built from variables and constants using operations such as addition, subtraction, multiplication and division. Examples: 3x + 5, 2a² − 4a + 1.
- Like terms have exactly the same variable part (same letters with the same powers). Only like terms can be combined when adding or subtracting expressions.
- To multiply two expressions, use the distributive property: multiply each term of the first expression by each term of the second, then combine like terms.
- Algebraic identities are equations that hold true for all values of the variables. Learning identities lets you expand or simplify expressions quickly.
- The three standard identities for squares are: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b².
- Identities can be verified by substituting numbers for the variables and checking that both sides give the same result.
- Algebra allows you to express general relationships, such as "the sum of three consecutive numbers is always divisible by 3" as n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1).
Formulas and facts to remember
- Identity / Rule: (a + b)² = a² + 2ab + b² · What it means: Square of a sum: the middle term is twice the product of a and b.
- Identity / Rule: (a − b)² = a² − 2ab + b² · What it means: Square of a difference: the middle term is negative.
- Identity / Rule: (a + b)(a − b) = a² − b² · What it means: Product of sum and difference gives difference of squares.
- Identity / Rule: Distributive property: a(b + c) = ab + ac · What it means: Multiply each term inside the bracket by the term outside.
- Identity / Rule: Combining like terms: 5x + 3x = 8x · What it means: Add or subtract coefficients; keep the variable part unchanged.
Worked examples
### Example 1 – Adding and subtracting expressions Simplify: (4x² + 3x − 7) + (2x² − 5x + 4)
Step 1: Write all terms together. 4x² + 3x − 7 + 2x² − 5x + 4
Step 2: Group like terms. (4x² + 2x²) + (3x − 5x) + (−7 + 4)
Step 3: Combine. 6x² − 2x − 3
Answer: 6x² − 2x − 3
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### Example 2 – Using an identity Expand (x + 4)² using the identity (a + b)² = a² + 2ab + b².
Here a = x and b = 4.
(x + 4)² = x² + 2 × x × 4 + 4² = x² + 8x + 16
Answer: x² + 8x + 16
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### Example 3 – Multiplying two binomials Find (2y + 3)(y − 5).
Step 1: Multiply each term of the first bracket by each term of the second. 2y × y = 2y² 2y × (−5) = −10y 3 × y = 3y 3 × (−5) = −15
Step 2: Add all four results. 2y² − 10y + 3y − 15
Step 3: Combine like terms. 2y² − 7y − 15
Answer: 2y² − 7y − 15
Common mistakes
- Forgetting the middle term when squaring: writing (a + b)² as a² + b² instead of a² + 2ab + b² → always include 2ab.
- Adding unlike terms: writing 5x + 3y as 8xy → keep unlike terms separate; they cannot be combined.
- Sign errors during subtraction: when subtracting (3x − 2), forgetting to change signs → distribute the minus: −(3x − 2) = −3x + 2.
- Mixing up (a − b)² and (a + b)(a − b): one gives a² − 2ab + b², the other gives a² − b² → check the pattern before applying.
- Dropping the variable part after multiplying: writing x × x as 2x instead of x² → remember x × x = x².
Quick revision
1. Like terms share the same variables raised to the same powers. 2. (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b². 3. (a + b)(a − b) = a² − b² (difference of squares). 4. Use the distributive property to multiply expressions: multiply every term by every term. 5. Combine like terms to simplify your final answer. 6. Substitute numbers to verify an identity works for all values.