What is Topic 6 in Post-UTME (Nigeria)?

Topic 6 is a topic in the Subject Combination syllabus of Post-UTME (Nigeria). It carries a weight of 3% in the exam. Our notes cover the key concepts, formulas, and problem-solving approaches you need to master this topic.

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Algebraic Expressions

Algebraic expressions are mathematical phrases that combine numbers, variables (letters), and operations. They form the foundation for solving equations and inequalities. For the Post-UTME examination, you must be able to simplify algebraic expressions, factorize them, expand brackets, and substitute values.

Understanding Algebraic Expressions

Key Terms

Variable: A letter that represents an unknown number (e.g., x, y, z)
Constant: A fixed number (e.g., 3, −5, π)
Term: A single number, variable, or product of numbers and variables (e.g., 3x², −7y, 5)
Coefficient: The numerical factor of a term (e.g., in 4x³, 4 is the coefficient)
Like Terms: Terms that have the same variable part (e.g., 3x² and 5x² are like terms)
Expression: A combination of terms connected by + or −

Simplifying Algebraic Expressions

Combining Like Terms

Only like terms can be combined.

Example: 3x² + 7x − 2x² + 4 − 5x + 3 = (3x² − 2x²) + (7x − 5x) + (4 + 3) = x² + 2x + 7

Removing Brackets (Expansion)

Use the distributive law: a(b + c) = ab + ac

Example: −2(x − 7) = −2x + 14 (careful with the negative sign!)

Expanding Double Brackets

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

Example: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15 Example: (2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12

Special Products

Difference of Two Squares: (a + b)(a − b) = a² − b²

(x + 3)(x − 3) = x² − 9
(2x + 1)(2x − 1) = 4x² − 1

Perfect Square Trinomials:

(a + b)² = a² + 2ab + b² → (x + 4)² = x² + 8x + 16
(a − b)² = a² − 2ab + b² → (3x − 2)² = 9x² − 12x + 4

Factorization

Factorization is the reverse of expansion — breaking down an expression into its factors.

Method 1: Common Factor

Example: 6x² + 9x = 3x(2x + 3)

Method 2: Difference of Two Squares

a² − b² = (a + b)(a − b) Example: x² − 16 = (x + 4)(x − 4); 4x² − 25 = (2x + 5)(2x − 5)

Method 3: Quadratic Factorization

For ax² + bx + c, find two numbers that multiply to give a×c and add to give b.

Example: x² + 7x + 12 → factors of 12 that add to 7: 3 and 4 = x² + 3x + 4x + 12 = x(x+3) + 4(x+3) = (x+3)(x+4)

Example: 2x² + 7x + 3 → a×c = 6; factors of 6 that add to 7: 6 and 1 = 2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)

Substitution

Substitution means replacing variables with given values and evaluating.

Example: If x = 3, y = −2, find 2x² − 3y + 5 = 2(9) − 3(−2) + 5 = 18 + 6 + 5 = 29

Example: If a = 2, b = −3, evaluate (a + b)² − (a − b)² Using identity: (a+b)² − (a−b)² = 4ab = 4(2)(−3) = −24

Algebraic Fractions

Simplify by factoring and canceling common factors.

Example: Simplify (x² − 9)/(x² − x − 6) = ((x+3)(x−3))/((x−3)(x+2)) = (x+3)/(x+2)

Adding fractions: Find LCD, express each with LCD, add numerators. Example: 1/(x+2) + 1/(x+3) = ((x+3)+(x+2))/[(x+2)(x+3)] = (2x+5)/[(x+2)(x+3)]

Key Post-UTME Exam Facts

(a+b)(a−b) = a²−b² — Difference of Two Squares
(x+p)(x+q) = x²+(p+q)x+pq — useful for quick expansion
Factorization: Common factor → Difference of squares → Quadratic splitting
⚡ Exam tip: Always check your factorization by expanding it back — if you get the original expression, it’s correct

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Expanding: Multiply each term inside by the term outside
(a+b)(a−b) = a²−b² — middle terms always cancel!
(a+b)² = a²+2ab+b²; (a−b)² = a²−2ab+b²
Factorization: Common factor first → Difference of squares → Quadratic splitting
Substitution: Replace variables; follow BODMAS; watch for negative signs
⚡ Exam tip: When expanding (a+b)(a−b), the middle terms always cancel — that’s the beauty of the difference of squares identity!

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Factorization by Grouping

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

Cubic Factorization

Example: x³ − 27 = (x − 3)(x² + 3x + 9) [using a³−b³ = (a−b)(a²+ab+b²)]

Algebraic Identities Table

Identity	Formula
(a+b)²	a² + 2ab + b²
(a−b)²	a² − 2ab + b²
(a+b)(a−b)	a² − b²
(a+b)³	a³ + 3a²b + 3ab² + b³
(a−b)³	a³ − 3a²b + 3ab² − b³
a³+b³	(a+b)(a²−ab+b²)
a³−b³	(a−b)(a²+ab+b²)

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Complex Algebraic Fractions

Example: Simplify (1/(x−1) − 1/(x+1)) ÷ (2/(x²−1))

Step 1: Combine subtraction: ((x+1)−(x−1))/[(x−1)(x+1)] = 2/[(x−1)(x+1)] Step 2: Divide by 2/(x²−1): 2/[(x−1)(x+1)] × [(x−1)(x+1)/2] = 1

Setting Up Algebraic Equations from Word Problems

Example: A rectangle’s length is 3 times its width. Perimeter = 80 cm. Find dimensions. Let width = w; Length = 3w; Perimeter = 2(3w+w) = 8w = 80 → w = 10 cm; Length = 30 cm

Example: Sum of three consecutive multiples of 4 is 180. Find them. Let: 4n, 4(n+1), 4(n+2); Sum: 4(3n+3) = 12(n+1) = 180 → n+1 = 15 → n = 14 Multiples: 56, 60, 64

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