Exercise 12.3 — Division of Expressions
Division of algebraic expressions.
What is Exercise 12.3 About?
Exercise 12.3 introduces the division of algebraic expressions — the reverse operation of multiplication. Just as you learned to multiply monomials and polynomials in Chapter 11, here you learn to split an expression by a divisor to find the quotient. This skill is essential for simplifying complex expressions, cancelling common factors, and preparing for polynomial long division in higher classes.
The exercise covers three types of division, all based on the idea of cancelling common factors in the numerator and denominator:
How Algebraic Division Works — The Key Concept
The law of exponents for division is the engine behind every problem in this exercise. When dividing two powers of the same variable, you subtract the exponent of the denominator from the exponent of the numerator:
xᵃ ÷ xᵇ = xᵃ⁻ᵇ (where a > b)For example, the PDF's introductory worked example demonstrates this clearly:
Question 1 — Divide Monomial by Monomial (Parts i–v)
Each part asks you to divide one monomial by another. The method: prime-factorise both the coefficient and the variable powers, then cancel identical factors top and bottom.
Coefficient: 72 ÷ 8 = 9 | a: 3−1=2 | b: 4−2=2 | c: 5−3=2= (1/5) × y²⁻¹ × z³⁻¹
= (1/5)yz²
= −(2×3) × l⁴⁻² × m³⁻² ← n² cancels
= −6 × l² × m
Question 2 — Divide a Polynomial by a Monomial (Parts i–vii)
When dividing a polynomial (multi-term expression) by a monomial, you divide each term of the polynomial separately by the monomial. Think of it as distributing the division across the addition and subtraction signs:
(A + B + C) ÷ M = (A/M) + (B/M) + (C/M)= −p² + 3pq + 2q²
= (2 × 2/3)abc + (2 × 4/3)bc
= (4/3)abc + (8/3)bc
Question 3 — Division When Expressions are in Factored Form
In these problems, both the dividend and divisor are given as products of factors (already in a partially factored form). The trick is to spot and cancel the common factor bracket directly — no need to expand first.
= 54(m+n) / 8
= 27(m+n) / 4
Cancel (x+4) from numerator & denominator
Question 4 — Factorise the Expression, then Divide
These are the most important problems in Exercise 12.3. Here, the numerator is a quadratic expression that you must first factorise into two brackets. Then the divisor cancels one of those brackets, leaving the other as the final answer. The factorisation identity used is:
x² + (a+b)x + ab = (x + a)(x + b)a² − b² = (a + b)(a − b) ← Difference of Squares[Difference of Squares: a²−b²=(a+b)(a−b)]
15lm × 2(p+q)(p−q) / 3l(p+q)
= (15×2m(p−q)) / 3 = 10m(p−q)
[16z²−9 = (4z)²−3² → difference of squares]
26z³ × 2(4z+3)(4z−3) / 13z²(4z−3)
= (26×2z(4z+3)) / 13 = 4z(4z+3)
All Answers at a Glance — Quick Reference Table
| Q | Division Problem | Method | Answer |
|---|---|---|---|
| Q1(i) | 48a³ ÷ 6a | Monomial ÷ Monomial | 8a² |
| Q1(ii) | 14x³ ÷ 42x² | Monomial ÷ Monomial | x/3 |
| Q1(iii) | 72a³b⁴c⁵ ÷ 8ab²c³ | Monomial ÷ Monomial | 9a²b²c² |
| Q1(iv) | 11xy²z³ ÷ 55xyz | Monomial ÷ Monomial | yz²/5 |
| Q1(v) | −54l⁴m³n² ÷ 9l²m²n² | Monomial ÷ Monomial | −6l²m |
| Q2(i) | (3x² − 2x) ÷ x | Polynomial ÷ Monomial | 3x − 2 |
| Q2(ii) | (5a³b − 7ab³) ÷ ab | Polynomial ÷ Monomial | 5a² − 7b² |
| Q2(iii) | (25x⁵ − 15x⁴) ÷ 5x³ | Polynomial ÷ Monomial | x(5x − 3) |
| Q2(iv) | (4l⁵ − 6l⁴ + 8l³) ÷ 2l² | Polynomial ÷ Monomial | l(2l² − 3l + 4) |
| Q2(v) | 15(a³b²c² − a²b³c² + a²b²c³) ÷ 3abc | Polynomial ÷ Monomial | 5abc(a − b + c) |
| Q2(vi) | (3p³ − 9p²q − 6pq²) ÷ (−3p) | Polynomial ÷ Monomial | −p² + 3pq + 2q² |
| Q2(vii) | (⅔a²b²c² + 4/3ab²c²) ÷ ½abc | Polynomial ÷ Monomial | (4/3)bc(a + 2) |
| Q3(i) | (49x − 63) ÷ 7 | Factored division | 7x − 9 |
| Q3(ii) | 12x(8x−20) ÷ 4(2x−5) | Factored division | 12x |
| Q3(iii) | 11a³b³(7c−35) ÷ 3a²b²(c−5) | Factored division | 77ab/3 |
| Q3(iv) | 54lmn(l+m)(m+n)(n+l) ÷ 8lmn(l+m)(n+l) | Factored division | 27(m+n)/4 |
| Q3(v) | 36(x+4)(x²+7x+10) ÷ 9(x+4) | Factored division | 4(x²+7x+10) |
| Q3(vi) | a(a+1)(a+2)(a+3) ÷ a(a+3) | Factored division | (a+1)(a+2) |
| Q4(i) | (x²+7x+12) ÷ (x+3) | Factorise → divide | x + 4 |
| Q4(ii) | (x²−8x+12) ÷ (x−6) | Factorise → divide | x − 2 |
| Q4(iii) | (p²+5p+4) ÷ (p+1) | Factorise → divide | p + 4 |
| Q4(iv) | 15ab(a²−7a+10) ÷ 3b(a−2) | Factorise → divide | 5a(a − 5) |
| Q4(v) | 15lm(2p²−2q²) ÷ 3l(p+q) | Diff. of squares | 10m(p − q) |
| Q4(vi) | 26z³(32z²−18) ÷ 13z²(4z−3) | Diff. of squares | 4z(4z + 3) |
Common Mistakes to Avoid in Exercise 12.3
- Subtracting instead of cancelling: When 6a ÷ 6a appears, students sometimes write 0 instead of 1. Dividing identical terms gives 1, not 0.
- Forgetting to divide every term: In polynomial ÷ monomial, every single term in the numerator must be divided. Missing even one term leads to a wrong answer.
- Ignoring the negative sign of the divisor: In Q2(vi), dividing by −3p flips all signs. Students often forget to flip the sign of every resulting term.
- Trying to cancel without factorising first: In Q4 problems, you cannot cancel (x+3) with x² — you must factorise x²+7x+12 into (x+3)(x+4) first.
- Wrong factor pairs for quadratics: When factorising x²+7x+12, find two numbers whose sum is 7 AND product is 12 simultaneously. A common error is to just find pairs that multiply to 12 without checking the sum.
- Forgetting the Difference of Squares formula: For Q4(v) and Q4(vi), recognising 2p²−2q² = 2(p+q)(p−q) and 16z²−9 = (4z+3)(4z−3) is essential. Always check if the expression is a difference of two perfect squares.
What This Exercise Prepares You For
Exercise 12.3 is the gateway to polynomial division across all levels. Once you are comfortable cancelling algebraic factors here, the next step is Exercise 12.4 — Finding and Correcting Errors, where you verify whether a given algebraic simplification is correct and identify where mistakes occurred.
In Class 9 and 10, you will use the same factorisation + cancellation technique in polynomial simplification and in quadratic equations. The difference-of-squares identity (a²−b²=(a+b)(a−b)) used in Q4(v) and Q4(vi) appears repeatedly in CBSE board questions across Class 9 and 10 as well.
For CBSE, Telangana, and Andhra Pradesh board examinations, algebraic division problems carry 2–3 marks each. Questions from Exercise 12.3 — especially the factorise-then-divide type (Q4) — are frequently asked in SA1 and SA2 assessments. Mastering every sub-question here gives you a reliable source of full marks.