Algebra Play
“Algebra Play” builds fluency with algebraic expressions. This Class 8 chapter covers terms and like terms, adding and subtracting expressions, the standard identities such as (a + b)², and how to factorise — writing an expression as a product — which is the reverse of expanding.
Learning objectives
- Identify terms, coefficients and like terms.
- Add and subtract algebraic expressions.
- Apply the standard algebraic identities.
- Factorise simple expressions.
Key concepts
Terms and like terms
An algebraic expression is built from terms, each a product of a number (the coefficient) and variables, such as 5x or −3xy. Like terms have exactly the same variable part (3x and 7x) and can be combined, while unlike terms (3x and 3y) cannot. Combining like terms simplifies an expression.
Adding and subtracting expressions
To add or subtract expressions, group like terms and combine their coefficients: (3x + 4) + (5x − 1) = 8x + 3. When subtracting, change the sign of every term being subtracted first: (5x − 1) − (2x + 3) = 5x − 1 − 2x − 3 = 3x − 4.
Algebraic identities
Identities are equalities true for all values of the variables. The key ones are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². They speed up squaring and multiplying, for example 102² = (100 + 2)² = 10000 + 400 + 4 = 10404.
Factorisation
Factorising means writing an expression as a product of factors — the reverse of expanding. We take out a common factor, as in 6x + 9 = 3(2x + 3), or use an identity, as in x² − 16 = (x + 4)(x − 4). Factorised forms make equations and simplifications easier.
Important formulas
Square of a sum
(a + b)² = a² + 2ab + b²
Square of a difference
(a − b)² = a² − 2ab + b²
Difference of squares
(a + b)(a − b) = a² − b²
Key definitions
- Term
- A product of a coefficient and variables within an expression.
- Like terms
- Terms having exactly the same variable part, which can be combined.
- Identity
- An equation true for all values of its variables.
- Factorisation
- Writing an expression as a product of its factors.
Solved examples
Q1. Simplify 4x + 7x − 2x.
Solution: All like terms: (4 + 7 − 2)x = 9x.
Q2. Use an identity to expand (x + 5)².
Solution: (x + 5)² = x² + 2·x·5 + 5² = x² + 10x + 25.
Q3. Factorise x² − 9.
Solution: This is a difference of squares: x² − 3² = (x + 3)(x − 3).
Common mistakes to avoid
- Combining unlike terms, e.g. writing 3x + 4y = 7xy.
- Forgetting the middle term: writing (a + b)² = a² + b².
- Not changing all signs when subtracting an expression.
- Leaving out a common factor when factorising.
Algebra Play — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
Which are like terms?
Practice questions
Short answer
What are like terms?
Terms with exactly the same variable part, such as 4x and 9x, which can be combined.
Write the identity for (a − b)².
(a − b)² = a² − 2ab + b².
Factorise 5x + 10.
5(x + 2).
Long answer
State the three standard identities and use one to evaluate 98² quickly.
The identities are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². To find 98², write 98 = 100 − 2 and use the second identity: 98² = (100 − 2)² = 100² − 2·100·2 + 2² = 10000 − 400 + 4 = 9604.
Explain factorisation and factorise (i) 6x + 9 and (ii) x² − 16.
Factorisation rewrites an expression as a product of factors, reversing expansion. (i) 6x + 9 has a common factor 3, so 6x + 9 = 3(2x + 3). (ii) x² − 16 is a difference of squares, x² − 4², which factorises using (a + b)(a − b) as (x + 4)(x − 4). Checking by expanding returns the original expression.
HOTS (Higher Order Thinking)
Without multiplying directly, find 53 × 47 using an identity.
Write as (50 + 3)(50 − 3) = 50² − 3² = 2500 − 9 = 2491, using the difference-of-squares identity.
Why is (a + b)² not equal to a² + b²?
Expanding (a + b)² gives a² + 2ab + b²; the extra term 2ab comes from multiplying the two parts together, so it cannot be dropped.
Quick revision
Revision notes
- Like terms share the variable part and combine; unlike terms do not.
- Subtracting an expression flips every sign inside.
- (a±b)² = a² ± 2ab + b²; (a+b)(a−b) = a² − b².
- Factorise by common factor or by an identity; it reverses expansion.
Key takeaways
- Identities turn hard squarings and products into quick steps.
- Factorisation is expansion run backwards.
- The middle term 2ab is the heart of a perfect square.
Frequently asked questions
What is the difference between an expression and an identity?
An expression is a combination of terms; an identity is an equation true for all values of its variables.
How do I factorise a difference of squares?
Write it as a² − b² and use (a + b)(a − b).
Does (a + b)² equal a² + b²?
No — it equals a² + 2ab + b²; the middle term 2ab must be included.