Algebraic Expressions

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

An algebraic expression combines constants, variables, and operation signs (+, −, ×, ÷, exponent) to represent a value, for example 5x² − 3x + 7. A term is a single product of a coefficient and variables raised to non-negative powers; like terms share the same variable part and can be combined, while unlike terms cannot.

Three identities carry roughly one-third of NAT-I marks in this topic:

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a − b)(a + b)

Identity	When to apply
Difference of squares	Two perfect squares separated by a minus sign
Perfect square trinomial	Three terms matching the (a ± b)² pattern
Common factor	Every term shares a number or variable

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Anatomy of an Expression

A monomial has one term (4x³), a binomial has two (3x + 5), and a trinomial has three (x² − 5x + 6). The degree equals the highest total exponent on the variables, and the leading coefficient is the number in front of the highest-degree term. The form ax² + bx + c = 0 is the standard quadratic form; here a ≠ 0 and solutions come from x = (-b ± √(b² − 4ac)) / 2a.

Simplification Workflow

Remove parentheses using the distributive property: a(b + c) = ab + ac.
Combine like terms by adding or subtracting coefficients only.
Apply exponent rules such as aᵐ · aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.

Worked Substitution

Evaluate 3x² − 2xy + y² when x = 4 and y = −1:

3(4)² = 3 · 16 = 48
−2(4)(−1) = 8
(−1)² = 1
Total: 48 + 8 + 1 = 57

Scoring tip: Write the substitution step on the rough sheet before adding — sign slips on the middle term cost 1–2 marks each in NAT-I scoring keys.

Operation	Rule to remember
Distribute a negative	Change the sign of every term inside the bracket
Combine like terms	Add coefficients, keep the variable part unchanged
Factor difference of squares	Take the square root of each square, wrap in `( )( )`

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Factoring Strategy for NAT-I Quadratics

Factor 6x² + 11x − 10 by the AC method: multiply a · c = 6 · (−10) = −60, then find two integers whose product is −60 and sum is 11, namely 15 and −4. Split the middle term: 6x² + 15x − 4x − 10. Group: 3x(2x + 5) − 2(2x + 5) = (2x + 5)(3x − 2). Verify by expanding back: 6x² − 4x + 15x − 10 = 6x² + 11x − 10. ✓

Word-Problem Translation Traps

“Exceeds by k” → add: x + k.
“Reduced to a fraction” → multiply: (p/q) · original.
“Successive percentages a% then b%” → multiply factors, do not add: x · (1 + a/100)(1 + b/100).

Common Mistakes

Dropping the sign when distributing −(a − b), writing −a − b instead of −a + b.
Mixing up (a − b)² = a² − 2ab + b² with a² − b² = (a − b)(a + b); the former has a middle term, the latter does not.
Combining 3x and 3x² because the coefficients match; only identical variable parts can be added.

Mistake	Correct form
`−(x − 3) = −x − 3`	`−(x − 3) = −x + 3`
`(a − b)² = a² − b²`	`(a − b)² = a² − 2ab + b²`
`3x + 3x² = 6x²`	`3x + 3x²` cannot be simplified

Practice Prompts

Factorize x² − 9 and evaluate at x = 7. (Expected: (x−3)(x+3), value 40.)
The length of a rectangle is 4 m more than its width; express its perimeter in terms of width w and find w when P = 36 m. (Expected: P = 4w + 8; w = 7 m.)

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Algebraic Expressions