Short Introduction
Exercise 4.5 focuses on simplifying rational algebraic expressions using factorisation techniques and algebraic identities. Students learn how to factor numerators and denominators, cancel common factors, and obtain simplified forms of algebraic fractions. These concepts are important for higher algebra and competitive examinations.
This complete solution guide by www.mymockmate.com provides detailed step-by-step explanations for every question of Exercise 4.5.
Quick Information Box
| Particular | Details |
|---|---|
| Class | 9 |
| Subject | Mathematics |
| Chapter | Exploring Algebraic Identities |
| Exercise | 4.5 |
| Topic | Simplification of Rational Expressions |
| Difficulty Level | Moderate |
| Exam Importance | High |
Concepts Used (Topics Covered)
✅ Rational Expressions
✅ Factorisation of Polynomials
✅ Difference of Squares
✅ Perfect Square Identities
✅ Common Factor Cancellation
✅ Algebraic Simplification
Important Formulas
Difference of Squares
a² − b² = (a+b)(a−b)
Perfect Square Identity
a² + 2ab + b² = (a+b)²
Perfect Square Identity
a² − 2ab + b² = (a−b)²
Middle Term Splitting
x² + (a+b)x + ab = (x+a)(x+b)
Rational Expression Rule
(a×b)/(a×c) = b/c, provided a ≠ 0
Question 1
Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:
(i)
(3p² − 3pq − 18q²)/(p² + 3pq − 10q²)
Answer
3(p − 3q)(p + 2q) / [(p + 5q)(p − 2q)]
(ii)
(n³ − 3n²m + 3nm² − m³)/(5m² − 10mn + 5n²)
Answer
(n − m)/5
(iii)
(w³ − v³ + x³ + 3wvx) / (w² + v² + x² − 2wv − 2vx + 2wx)
Answer
w + x − v
(iv)
(4y² − 20yz + 25z²)/(25z² − 4y²)
Answer
(5z − 2y)/(5z + 2y)
(v)
[(x²+x−6)(x²−7x+12)] / [(x²−6x+8)(x²−9)]
Answer
1
(vi)
(p⁴ − 16)/(p² − 4p + 4)
Answer
[(p+2)(p²+4)]/(p−2)
Final Answers Summary
| Question | Answer |
|---|---|
| (i) | 3(p−3q)(p+2q)/[(p+5q)(p−2q)] |
| (ii) | (n−m)/5 |
| (iii) | w+x−v |
| (iv) | (5z−2y)/(5z+2y) |
| (v) | 1 |
| (vi) | [(p+2)(p²+4)]/(p−2) |
Common Mistakes
❌ Forgetting to factor completely before simplifying.
❌ Cancelling terms instead of factors.
❌ Sign errors in difference of squares.
❌ Incorrect factorisation of quadratic expressions.
❌ Ignoring denominator restrictions.
Exam Tips
✅ Always factor numerator and denominator completely.
✅ Cancel only common factors.
✅ Check signs carefully.
✅ Write intermediate factorisation steps.
✅ Verify the final answer by substitution if required.
Practice MCQs
1. Factor x²−25
A. (x−5)²
B. (x+5)(x−5)
C. (x+25)
D. None
✅ Answer: B
2. Simplify (x²−9)/(x−3)
A. x−3
B. x+3
C. x²+3
D. 1
✅ Answer: B
3. Which identity is used in Question (iv)?
A. (a+b)²
B. (a−b)²
C. a²−b²
D. Both B and C
✅ Answer: D
4. Simplify (a²−b²)/(a−b)
A. a−b
B. a+b
C. a²+b²
D. 1
✅ Answer: B
5. Factor p²−4p+4
A. (p−2)²
B. (p+2)²
C. (p−4)(p+1)
D. None
✅ Answer: A
FAQ Section
Q1. What is a rational expression?
An algebraic fraction containing variables in numerator and denominator.
Q2. Why do we factor expressions before simplifying?
Factorisation helps identify common factors that can be cancelled.
Q3. Can terms be cancelled directly?
No. Only common factors can be cancelled.
Q4. Which identity is most frequently used in Exercise 4.5?
Difference of squares:
a²−b²=(a+b)(a−b)
Q5. Why are denominator restrictions important?
Because division by zero is undefined.
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