Class 9 Maths Chapter 2 Polynomials – Exercise 2.5 NCERT Solutions

Introduction

Exercise 2.5 focuses on the factorisation of polynomials using identities and splitting the middle term. Students learn how to apply standard algebraic identities and factorisation techniques to simplify polynomial expressions. This exercise is crucial for building a strong foundation in algebra.

Key Concepts

1. Standard Identities:

$$(x+y{)}^2=x^2+2xy+y^2$$

$$(x-y{)}^2=x^2-2xy+y^2$$

$$(x+y)(x-y)=x^2-y^2$$

2. Factorisation by Middle Term Splitting: For quadratic $a{x}^2+bx+c$, split $b$ into two terms whose product = $ac$.

3. Common Factor Method: Take out common terms to simplify polynomial.

Common Mistakes

• Forgetting to apply correct identity.

• Incorrectly splitting middle term.

• Missing negative signs in factorisation.

• Not checking factors by multiplication.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Factorise: $x^2+5x+6$.

$$x^2+5x+6=x^2+2x+3x+6=(x+2)(x+3)$$

Q2. Factorise: $x^2-7x+10$.

$$x^2-7x+10=x^2-5x-2x+10=(x-5)(x-2)$$

Q3. Factorise: $x^2+11x+24$.

$$x^2+11x+24=x^2+8x+3x+24=(x+8)(x+3)$$

Q4. Factorise: $x^2-13x+42$.

$$x^2-13x+42=x^2-7x-6x+42=(x-7)(x-6)$$

Q5. Factorise: $x^2+12x+35$.

$$x^2+12x+35=x^2+7x+5x+35=(x+7)(x+5)$$

Q6. Factorise: $x^2-15x+50$.

$$x^2-15x+50=x^2-10x-5x+50=(x-10)(x-5)$$

Q7. Factorise: $x^2+9x+20$.

$$x^2+9x+20=x^2+5x+4x+20=(x+5)(x+4)$$

Q8. Factorise: $x^2-10x+21$.

$$x^2-10x+21=x^2-7x-3x+21=(x-7)(x-3)$$

Q9. Factorise: $x^2+14x+45$.

$$x^2+14x+45=x^2+9x+5x+45=(x+9)(x+5)$$

Q10. Factorise: $x^2-8x+15$.

$$x^2-8x+15=x^2-5x-3x+15=(x-5)(x-3)$$

FAQs (10)

FAQ1. What is factorisation? Breaking polynomial into product of simpler polynomials.

FAQ2. What is middle term splitting? Splitting middle term to factorise quadratic.

FAQ3. What are standard identities? Formulas like $(x+y{)}^2$, $(x-y{)}^2$, $(x+y)(x-y)$.

FAQ4. Why check factors by multiplication? To verify correctness.

FAQ5. What is common factor method? Taking out common terms.

FAQ6. What is quadratic polynomial? Polynomial of degree 2.

FAQ7. Can cubic polynomials be factorised? Yes, using identities or grouping.

FAQ8. Why is factorisation important? Used in solving equations.

FAQ9. What is difference of squares identity? $(x+y)(x-y)=x^2-y^2$.

FAQ10. Why is Exercise 2.5 important? It builds foundation for solving polynomial equations.

Conclusion

Exercise 2.5 covers factorisation of polynomials with solved examples and FAQs. Mastering these problems helps students in algebra, equations, and higher mathematics.

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