Class 9 Exam  >  Class 9 Notes  >  RD Sharma Solutions for Class 9 Mathematics  >  RD Sharma Solutions -Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q 1 . p3+27

SOLUTION :

= p3+33                                       ∵[a3+b3=(a+b)(a2−ab+b2)]

= (p + 3)(p² – 3p – 9)

∴ p3+27 =(p + 3)(p² – 3p – 9)

 

Q 2 . y3+125

SOLUTION :

=  y3+5                                      ∵[a3+b3=(a+b)(a2−ab+b2)]

= (y+5)(y2−5y+52)

= (y+5)(y2−5y+25)

∴ y3+125 = (y+5)(y2−5y+25)

 

Q 3 . 1−27a3

SOLUTION :

= (1)3−(3a)3

= (1−3a)(12+1×3a+(3a)2)                  ∵[a3−b3=(a−b)(a2+ab+b2)]

= (1−3a)(12+3a+9a2)

∴ 1−27a= (1−3a)(12+3a+9a2)

 

Q 4 . 8x3y3+27a3

SOLUTION :

= (2xy)3+(3a)3

= (2xy+3a)((2xy)2−2xy×3a+(3a)2)                        ∵[a3+b3=(a+b)(a2−ab+b2)]

= (2xy+3a)(4x2y2−6xya+9a2)

∴ 8x3y3+27a3 = (2xy+3a)(4x2y2−6xya+9a2)

 

Q 5 . 64a3−b3

SOLUTION :

= (4a)3−b3

= (4a−b)((4a)2+4a×b+b2)                                 ∵[a3−b3=(a−b)(a2+ab+b2)]

=(4a−b)(16a2+4ab+b2)

∴  64a3−b3 =(4a−b)(16a2+4ab+b2)

 

 

Q 6 .  Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION :

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics         ∵ [x3−y3=(x−y)(x2+xy+y2)]

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics 

 

Q 7 . 10x4y−10xy4

SOLUTION  :

= 10xy(x3−y3)

= 10xy(x−y)(x2+xy+y2)                          ∵[x3−y3=(x−y)(x2+xy+y2)]

∴ 10x4y−10xy4  = 10xy(x−y)(x2+xy+y2)

 

Q 8 . 54x6y+2x3y4

SOLUTION  :

= 2x3y(27x3+y3)

= 2x3y((3x)3+y3)

= 2x3y(3x+y)((3x)2−3x×y+y2)                  ∵[a3+b3=(a+b)(a2−ab+b2)]

=2x3y(3x+y)(9x2−3xy+y2)

∴ 54x6y+2x3y4 =2x3y(3x+y)(9x2−3xy+y2)

 

Q 9 . 32a3+108b3

SOLUTION  :

= 4(8a3+27b3)

= 4((2a)3+(3b)3)

= 4[(2a+3b)((2a)2−2a×3b+(3b)2)]                                                ∵[a3+b3=(a+b)(a2−ab+b2)]

=4(2a+3b)(4a2−6ab+9b2)

∴ 32a3+108b3 =4(2a+3b)(4a2−6ab+9b2)

 

Q 10 . (a−2b)3−512b3

SOLUTION  :

= (a−2b)3−(8b)3

= (a−2b−8b)((a−2b)2+(a−2b)8b+(8b)2)                                    ∵[a3−b3=(a−b)(a2+ab+b2)]

=(a−10b)(a2+4b2−4ab+8ab−16b2+64b2)

=(a−10b)(a2+52b2+4ab)

∴  (a−2b)3−512b3 =(a−10b)(a2+52b2+4ab)

 

Q 11 . (a+b)3−8(a−b)3

SOLUTION  :

= (a+b)3−[2(a−b)]3

= (a+b)3−[2a−2b]3

= (a+b−(2a−2b))((a+b)2+(a+b)(2a−2b)+(2a−2b)2)                                                                      ∵[a3−b3=(a−b)(a2+ab+b2)]

=(a+b−2a+2b)(a2+b2+2ab+(a+b)(2a−2b)+(2a−2b)2)

=(a+b−2a+2b)(a2+b2+2ab+2a2−2ab+2ab−2b2+(2a−2b)2)

=(3b−a)(3a2+2ab−b2+(2a−2b)2)

=(3b−a)(3a2+2ab−b2+4a2+4b2−8ab)

=(3b−a)(3a2+4a2−b2+4b2−8ab+2ab)

=(3b−a)(7a2+3b2−6ab)

∴  (a+b)3−8(a−b)3 =(3b−a)(7a2+3b2−6ab)

 

Q 12 . (x+2)3+(x−2)3

SOLUTION  :

= (x+2+x−2)((x+2)2−(x+2)(x−2)+(x−2)2)                                               ∵[a3+b3=(a+b)(a2−ab+b2)]

=2x(x2+4x+4−(x+2)(x−2)+x2−4x+4)

=2x(2x2+8−(x2−22))                           [∵(a+b)(a−b)=a2−b2]

=2x(2x2+8−x2+4)

=2x(x2+12)

∴ (x+2)3+(x−2)3 =2x(x2+12)

 

Q 13 .  8x2y3−x5

SOLUTION  :

= x2((2y)3−x3)

= x2(2y−x)((2y)2+2y×x+x2)     [∵x3−y3=(x−y)(x2+xy+y2)]

= x2(2y−x)(4y2+2xy+x2)

∴ 8x2y3−x= x2(2y−x)(4y2+2xy+x2)

 

Q  14 . 1029 – 3x3

SOLUTION  :

= 3(343−x3)

= 3((7)3−x3)

= 3(7−x)(72+7x+x2)                                    [∵a3−b3=(a−b)(a2+ab+b2)]

=3(7−x)(49+7x+x2)

∴ 1029 – 3x3 =3(7−x)(49+7x+x2)

 

Q 15 . x6+y6

SOLUTION  :

= (x2)3+(y2)3

= (x2+y2)((x2)2−x2y2+(y2)2)

= (x2+y2)(x4−x2y2+y4)                                                [∵a3+b3=(a+b)(a2−ab+b2)]

∴ x6+y6 = (x2+y2)(x4−x2y2+y4)

 

Q 16 . x3y3+1

SOLUTION  :

= (xy)3+13

= (xy+1)((xy)2+xy+12)                                                [∵x3+y3=(x+y)(x2−xy+y2)]

=(xy+1)(x2y2−xy+1)

∴ x3y3+1 = (xy+1)(x2y2−xy+1)

 

Q 17 . x4y4−xy

SOLUTION  :

= xy(x3y3−1)

= xy((xy)3−13)

= xy(xy−1)((xy)2+xy×1+12)                                             ∵[x3−y3=(x−y)(x2+xy+y2)]

=xy(xy−1)(x2y2+xy+1)

∴ x4y4−xy = xy(xy−1)(x2y2+xy+1)

 

Q 18 . a12+b12

SOLUTION  :

= (a4)3+(b4)3

= (a4+b4)((a4)2−a4×b4+(b4)2)                                                ∵[a3+b3=(a+b)(a2−ab+b2)]

=(a4+b4)(a8−a4b4+b8)

∴ a12+b12 = (a4+b4)(a8−a4b4+b8)

 

Q 19 . x3+6x2+12x+16

SOLUTION  :

= x3+6x2+12x+8+8

= x3+3×x2×2+3×x×22+23+8

= (x+2)3+8                                              [∵a3+3a2b+3ab2+b3=(a+b)3]

= (x+2)3+23

= (x+2+2)((x+2)2−2(x+2)+22)                            ∵[a3+b3=(a+b)(a2−ab+b2)]

=(x+2+2)(x2+4+4x−2x−4+4)                       [∵(a+b)2=a2+b2+2ab]

=(x+4)(x2+4+2x)

∴ x3+6x2+12x+16 = (x+4)(x2+4+2x)

 

Q 20 . a3+b3+a+b

SOLUTION  :

= (a3+b3)+1(a+b)

= (a+b)(a2−ab+b2)+1(a+b)                                         [∵a3+b3=(a+b)(a2−ab+b2)]

=(a+b)(a2−ab+b2+1)

∴ a3+b3+a+b = (a+b)(a2−ab+b2+1)

 

Q 21 . Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 [∵a3−b3=(a−b)(a2+ab+b2)]

  Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 

Q 22 . a3+3a2b+3ab2+b3−8

SOLUTION  :

= (a+b)3−8                                                                [∵a3+3a2b+3ab2+b3=(a+b)3]

=(a+b)3−23

=(a+b−2)((a+b)2+(a+b)×2+22)

= (a + b – 2)(a² + 2ab + b² + 2a + 2b + 4)

∴ a3+3a2b+3ab2+b3−8 = (a + b – 2)(a² + 2ab + b² + 2a + 2b + 4)

 

Q 23 . 8a3−b3−4ax+2bx

SOLUTION  :

= (2a)3−b3−2x(2a−b)

= (2a−b)((2a)2+2a×b+b2)−2x(2a−b)                      [∵a3−b3=(a−b)(a2+ab+b2)]

=(2a−b)(4a2+2ab+b2−2x)

∴ 8a3−b3−4ax+2bx= (2a−b)(4a2+2ab+b2−2x)

 

Q 24 . i .Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics             ∵[a3+b3=(a+b)(a2−ab+b2)]

=(173+127)

=300

 

Q 24 . ii .Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics                [∵a3−b3=(a−b)(a2+ab+b2)]

=(1.2−0.2)

=1.0

 

Q 24 . iii   Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics           [∵a3−b3=(a−b)(a2+ab+b2)]

= (155 – 55)

= 100

 

 

The document Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.
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FAQs on Ex-5.2, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions - RD Sharma Solutions for Class 9 Mathematics

1. What is the importance of factorization of algebraic expressions in class 9 mathematics?
Ans. Factorization of algebraic expressions is important in class 9 mathematics as it helps in simplifying complex expressions and solving equations. It also enables us to find common factors and identify patterns in expressions, making it easier to manipulate and solve equations.
2. How can factorization of algebraic expressions be helpful in solving real-life problems?
Ans. Factorization of algebraic expressions is helpful in solving real-life problems by providing a systematic approach to simplifying and solving equations. It allows us to break down complex problems into simpler forms, making them easier to analyze and find solutions. This skill is useful in various fields such as engineering, physics, and finance, where equations and expressions are frequently encountered.
3. Can you explain the process of factorization of algebraic expressions in class 9 mathematics?
Ans. The process of factorization involves breaking down an algebraic expression into its simplest form by identifying common factors and applying relevant algebraic techniques. This can be done through methods such as finding the greatest common factor, using the distributive property, or applying specific factorization formulas. It is important to understand the properties and rules of algebra to effectively factorize expressions.
4. What are some common factorization techniques used in class 9 mathematics?
Ans. Some common factorization techniques used in class 9 mathematics include: - Finding the greatest common factor (GCF): This involves identifying the highest common factor of all terms in the expression and factoring it out. - Using the distributive property: This technique involves distributing a common factor to all terms in the expression, thereby simplifying it. - Applying specific factorization formulas: There are specific formulas for factorizing expressions such as the difference of squares, perfect square trinomials, or the sum and difference of cubes.
5. Can you provide an example of factorization of an algebraic expression in class 9 mathematics?
Ans. Sure! Let's consider the expression 3x^2 + 6xy. We can factor out the greatest common factor, which is 3, from both terms, resulting in 3(x^2 + 2xy). This is an example of factorization where we simplify the expression by finding the common factor and factoring it out.
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