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RS Aggarwal Solutions: Factorisation of Polynomials- 1 | Mathematics (Maths) Class 9 PDF Download

RS Aggarwal Solutions: Exercise 3A - Factorisation of Polynomials



Q.1. Factorize: 9x2 + 12xy
Ans. We have: 9x2+12xy
= 3x(3x+4y)

Q.2. Factorize: 18x2y − 24xyz
Ans. 
We have: 18x2y-24xyz
= 6xy(3y-4z)

Q.3. Factorize: 27a3b3 − 45a4b2

Ans. We have: 27a3b3-45a4b2
= 9a3b2(3b-5a)

Q.4. Factorize: 2a(x + y) − 3b(x + y)
Ans.
We have: 2a(x+y)-3b(x+y)
=(x+y)(2a-3b)


Q.5. Factorize: 2x(p2 + q2) + 4y(p2 + q2)
Ans. 
We have: 2x(p2+q2)+4y(p2+q2)
= 2[x(p2+q2)+2y(p2+q2)]
= 2(p2+q2)(x+2y)

Q.6. Factorize: x(a − 5) + y(5 − a)
Ans.
We have:
x(a-5)+y(5-a)
= x(a-5)-y(a-5)
=(a-5)(x-y)


Q.7. Factorize: 4(a + b) − 6(a + b)2
Ans.
We have: 4(a+b)-6(a+b)2
=2(a+b)[2-3(a+b)]
=2(a+b)(2-3a-3b)


Q.8. Factorize: 8(3a − 2b)2 − 10(3a − 2b)
Ans.
We have: 8(3a-2b)2-10(3a-2b)=2(3a-2b)[4(3a-2b)-5]
=2(3a-2b)(12a-8b-5)

Q.9. Factorize: x(x + y)3 − 3x2y(x + y)
Ans.
We have: x(x+y)3-3x2y(x+y)=x(x+y)[(x+y)2-3xy]
=x(x+y)[x2+y2+2xy-3xy]
=x(x+y)(x2+y2-xy)

Q.10. Factorize: x3 + 2x2 + 5x + 10
Ans.
We have: x3+2x2+5x+10=(x3+2x2)+(5x+10)
=x2(x+2)+5(x+2)
=(x+2)(x2+5)

Q.11. Factorize: x2 + xy − 2xz − 2yz
Ans. 
We have: x2+xy-2xz-2yz=(x2+xy)-(2xz+2yz)
=x(x+y)-2z(x+y)
=(x+y)(x-2z)

Q.12. Factorize: a3b − a2b + 5ab − 5b
Ans.
We have: a3b-a2b+5ab-5b=b(a3-a2+5a-5)
=b[(a3-a2)+(5a-5)]
=b[a2(a-1)+5(a-1)]
=b(a-1)(a2+5)

Q.13. Factorize: 8 − 4a − 2a3 + a4
Ans.
We have: 8-4a-2a3+a4= (8-4a)-(2a3-a4)
= 4(2-a)- a3(2-a)
= (2-a) (4 - a3)

Q.14. Factorize: x3 − 2x2y + 3xy2 − 6y3
Ans.
We have: x3-2x2y+3xy2-6y3
=(x3-2x2y)+(3xy2-6y3)
=x2(x-2y)+3y2(x-2y)
=(x-2y)(x2+3y2)

Q.15. Factorize: px − 5q + pq − 5x
Ans. 
We have: px-5q+pq-5x
=(px-5x)+(pq-5q)
=x(p-5)+q(p-5)
=(p-5)(x+q)

Q.16. Factorize: x2 + y − xy − x
Ans.
We have: x2+y-xy-x=(x2-xy)-(x-y)
=x(x-y)-1(x-y)
=(x-y)(x-1)

Q.17. Factorize: (3a − 1)2 − 6a + 2
Ans. 
We have: (3a-1)2-6a+2=(3a-1)2-2(3a-1)
=(3a-1)[(3a-1)-2]
=(3a-1)(3a-1-2)
=(3a-1)(3a-3)
=3(3a-1)(a-1)

Q.18. Factorize: (2x − 3)2 − 8x + 12
Ans.
We have: (2x-3)2-8x+12
=(2x-3)2-4(2x-3)
=(2x-3)[(2x-3)-4]
=(2x-3)(2x-3-4)
=(2x-3)(2x-7)

Q.19. Factorize: a3 + a − 3a2 − 3
Ans. 
We have: a3+a-3a2-3
=(a3-3a2)+(a-3)
=a2(a-3)+1(a-3)
=(a-3)(a2+1)

Q.20. Factorize: 3ax − 6ay − 8by + 4bx
Ans.
We have: 3ax-6ay-8by+4bx
=(3ax-6ay)+(4bx-8by)
=3a(x-2y)+4b(x-2y)
=(x-2y)(3a+4b)

Q.21. Factorize: abx2 + a2x + b2x + ab
Ans.
We have: abx2+a2x+b2x+ab=(abx2+b2x)+(a2x+ab)
=bx(ax+b)+a(ax+b)=(ax+b)(bx+a)

Q.22. Factorize: x3 − x2 + ax + x − a − 1
Ans. 
We have: x3-x2+ax+x-a-1
=(x3-x2)+(ax-a)+(x-1)
=x2(x-1)+a(x-1)+1(x-1)
=(x-1)(x2+a+1)

Q.23. Factorize: 2x + 4y − 8xy − 1
Ans. 
We have: 2x+4y−8xy−1=(2x−8xy)−(1−4y)
=2x(1−4y)−1(1−4y)
=(1−4y)(2x−1)

Q.24. Factorize: ab(x2 + y2) − xy(a2 + b2)
Ans.
We have: ab(x2+y2)−xy(a2+b2)
=abx2+aby2−a2xy−b2xy
=(abx2−a2xy)−(b2xy−aby2)
=ax(bx−ay)−by(bx−ay)
=(bx−ay)(ax−by)

Q.25. Factorize: a2 + ab(b + 1) + b3
Ans. 
We have: a2+ab(b+1)+b3=a2+ab2+ab+b3
=(a2+ab2)+(ab+b3)
=a(a+b2)+b(a+b2)
=(a+b2)(a+b)

Q.26. Factorize: a3 + ab(1 − 2a) − 2b2
Ans. 
We have: a3+ab(1−2a)−2b2=a3+ab−2a2b−2b2
=(a3−2a2b)+(ab−2b2)
=a2(a−2b)+b(a−2b)
=(a−2b)(a2+b)

Q.27. Factorize: 2a2 + bc − 2ab − ac2
Ans.
We have: 2a2+bc−2ab−ac=(2a2−2ab)−(ac−bc)
=2a(a−b)−c(a−b)
=(a−b)(2a−c)

Q.28. Factorize: (ax + by)2 + (bx − ay)2
Ans.
We have: (ax+by)2+(bx−ay)2
=[(ax)2+2×ax×by+(by)2]+[(bx)2−2×bx×ay+(ay)2]
=a2x2+2abxy+b2y2+b2x2−2abxy+a2y2
=a2x2+b2y2+b2x2+a2y2
=(a2x2+b2x2)+(a2y2+b2y2)
=x2(a2+b2)+y2(a2+b2)
=(a2+b2)(x2+y2)

Q.29. Factorize: a(a + b − c) − bc
Ans.
We have: a(a+b−c)−bc
=a2+ab−ac−bc
=(a- ac) + (ab-bc)
=a(a−c)+b(a−c)
=(a−c)(a+b)

Q.30. Factorize: a(a − 2b − c) + 2bc
Ans. 
We have: a(a−2b−c)+2bc=a2−2ab−ac+2bc
=(a2−2ab)−(ac−2bc)
=a(a−2b)−c(a−2b)
=(a−2b)(a−c)

Q.31. Factorize: a2x2 + (ax2 + 1)x + a
Ans. 
We have: a2x2+(ax2+1)x+a=(ax2+1)x+(a2x2+a)
=x(ax2+1)+a(ax2+1)
=(ax2+1)(x+a)

Q.32. Factorize: ab(x2 + 1) + x(a2 + b2)
Ans.
We have: ab(x2+1)+x(a2+b2)
=abx2+ab+a2x+b2x
=(abx2+a2x)+(b2x+ab)
=ax(bx+a)+b(bx+a)
=(bx+a)(ax+b)

Q.33. Factorize: x2 − (a + b)x + ab
Ans. 
We have: x2−(a+b)x+ab
=x2−ax−bx+ab
=x(x−a)−b(x−a)
=(x−a)(x−b)

Q.34. Factorize: x2+1/x2−2−3x+3/x
Ans.
We have: x2+1/x2−2−3x+3x/4
= x2−2+1/x2−3x+3/x
=(x)2−2×x×1/x+(1/x)2−3(x−1/x)
=(x−1/x)2−3(x−1/x)
=(x−1/x)(x−1/x−3)

The document RS Aggarwal Solutions: Factorisation of Polynomials- 1 | Mathematics (Maths) Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on RS Aggarwal Solutions: Factorisation of Polynomials- 1 - Mathematics (Maths) Class 9

1. What is factorisation of polynomials?
Ans. Factorisation of polynomials is the process of expressing a polynomial as a product of its factors. It involves breaking down a polynomial into simpler and more manageable factors.
2. Why is factorisation of polynomials important?
Ans. Factorisation of polynomials is important because it helps in simplifying complex expressions, finding the roots of equations, solving equations, and understanding the behavior of polynomials. It also plays a crucial role in various mathematical applications.
3. How do you factorise a quadratic polynomial?
Ans. To factorise a quadratic polynomial, we can use various methods such as the trial and error method, the method of splitting the middle term, or by using the formula for the factorisation of a quadratic equation. These methods help in expressing the quadratic polynomial as a product of two linear factors.
4. Can all polynomials be factorised?
Ans. No, not all polynomials can be factorised. While some polynomials can be easily factorised, there are certain polynomials, especially higher degree polynomials, that may not have any factors or may have factors that are complex or irrational numbers. In such cases, the polynomial is said to be irreducible.
5. What is the importance of practicing factorisation of polynomials?
Ans. Practicing factorisation of polynomials helps in strengthening the understanding of algebraic concepts, improving problem-solving skills, and preparing for higher-level mathematics. It also helps in identifying patterns and relationships between different polynomial expressions, which can be useful in various mathematical applications and real-life scenarios.
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