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Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q 1 . a3+8b3+64c3−24abc

SOLUTION  :

= (a)3+(2b)3+(4c)3−3×a×2b×4c

= (a+2b+4c)(a2+(2b)2+(4c)2−a×2b−2b×4c−4c×a)                                   [∵a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)]

= (a+2b+4c)(a2+4b2+16c2−2ab−8bc−4ac)

∴   a3+8b3+64c3−24abc = (a+2b+4c)(a2+4b2+16c2−2ab−8bc−4ac)

 

Q 2 . x3−8y3+27z3+18xyz

SOLUTION  :

= x3−(2y)3+(3z)3−3×x×(−2y)(3z)

= (x+(−2y)+3z)(x2+(−2y)2+(3z)2−x(−2y)−(−2y)(3z)−3z(x))                            [∵a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)]

=(x+(−2y)+3z)(x2+4y2+9z2+2xy+6yz−3zx)

∴   x3−8y3+27z3+18xyz =(x+(−2y)+3z)(x2+4y2+9z2+2xy+6yz−3zx)

 

Q 3 . Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematicsx3−y3+125z3+5xyz

SOLUTION  :

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics+(−y)3+(5z)3−3×Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics(−y)(5z)

= (Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics+(−y)+5z)((Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics)2+(−y)2+(5z)2Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics(−y)−(−y)5z−5z(Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics))

= (Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics−y+5z)Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

∴   Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematicsx3−y3+125z3+5xyz = (Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics−y+5z)Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 

Q 4 . 8x3+27y3−216z3+108xyz

SOLUTION  :

= (2x)3+(3y)3+(−6y)3−3(2x)(3y)(−6z)

= (2x+3y+(−6z))((2x)2+(3y)2+(−6z)2−2x×3y−3y(−6z)−(−6z)2x)

= (2x+3y+(−6z))(4x2+9y2+36z2−6xy+18yz+12zx)

∴   8x3+27y3−216z3+108xyz = (2x+3y+(−6z))(4x2+9y2+36z2−6xy+18yz+12zx)

 

Q 5 . 125+8x3−27y3+90xy

SOLUTION  :

= (5)3+(2x)3+(−3y)3−3×5×2x×(−3y)

= (5+2x+(−3y))(52+(2x)2+(−3y)2−5(2x)−2x(−3y)−(−3y)5)

= (5+2x−3y)(25+4x2+9y2−10x+6xy+15y)

∴   125+8x3−27y3+90xy = (5+2x−3y)(25+4x2+9y2−10x+6xy+15y)

 

Q 6 . (3x−2y)3+(2y−4z)3+(4z−3x)3

SOLUTION  :

Let  (3x−2y) = a , (2y−4z) = b , (4z−3x) = c

∴ a+b+c=3x−2y+2y−4z+4z−3x=0

∵ a+b+c=0                             ∴ a3+b3+c3=3abc

= 3(3x−2y)(2y−4z)(4z−3x)

∴   (3x−2y)3+(2y−4z)3+(4z−3x)3  = 3(3x−2y)(2y−4z)(4z−3x)

 

Q 7 .  (2x−3y)3+(4z−2x)3+(3y−4z)3

SOLUTION  :

Let 2x – 3y = a , 4z – 2x = b , 3y – 4z = c

∴ a+b+c=2x−3y+4z−2x+3y−4z=0

∵ a+b+c=0            ∴   a3+b3+c3=3abc

= 3(2x−3y)(4z−2x)(3y−4z)

∴   (2x−3y)3+(4z−2x)3+(3y−4z)3 = 3(2x−3y)(4z−2x)(3y−4z)

 

Q 8 . Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Let Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics  Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

a+b+c= Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

a+b+c=  Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

a+b+c=  Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

a+b+c= Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

a+b+c= 0

 

∵a+b+c=0                    ∴   a3+b3+c3=3abc

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 

Q 9 . (a−3b)3+(3b−c)3+(c−a)3

SOLUTION  :

Let  a – 3b = x , 3b – c = y , c – a = z

x+y+z=a−3b+3b−c+c−a=0

(∵ x+y+z=0)                      ∴  x3+y3+z3=3xyz

= 3(a−3b)(3b−c)(c−a)

∴   (a−3b)3+(3b−c)3+(c−a)3 = 3(a−3b)(3b−c)(c−a)

 

Q 10 . Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 

Q 11 .Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 

Q 12 . 8x3−125y3+216+180xy

SOLUTION  :

= (2x)3+(−5y)3+63−3×(2x)(−5y)(6)

= (2x+(−5y)+6)((2x)2+(−5y)2+62−2x×(−5y)−(−5y)6−6(2x))

= (2x−5y+6)(4x2+25y2+36+10xy+30y−12x)

∴   8x3−125y3+216+180xy = (2x−5y+6)(4x2+25y2+36+10xy+30y−12x)

 

Q 13 .  Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

SOLUTION  :

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Ex-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 MathematicsEx-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

 

 

 

Q 14 . Find the value of   x3+y3−12xy+64 when x + y = -4.

SOLUTION  :

= x3+y3+64−12xy

= x3+y3+43−3(x)(y)(4)

= (x+y+4)(x2+y2+42−xy−y×4−4×x)

= (−4+4)(x2+y2+16−xy−4y−4x)                            [∵x+y=−4]

=0

∴   x3+y3−12xy+64 = 0

 

Q 15 . MULTIPLY :

(i) . x2+y2+z2−xy+xz+yzbyx+y−z

SOLUTION  :

= (x2+y2+z2−xy+xz+yz)(x+y−z)

= x3+y3+z3−3xyz

(ii) . x2+4y2+z2+2xy+xz−2yzbyx−2y−z

SOLUTION  :

x2+(−2y)2+(−z)2−(−2y)(−z)−(−z)(x) = x3+(−2y)3+(−z)3−3x(−2y)(−z)

⇒x2+4y2+z2+2xy−2yz+zx=x3−8y3−z3−6xyz

(iii) . x2+4y2+2xy−3x+6y+9  by  (x−2y+3)

SOLUTION  :

(x)2+(−2y)2+(3)2−(x)(−2y)−(−2y)(3)−3(x)= (x)3+(−2y)3+33−3(x)(−2y)(3)

⇒x2+4y2+9+2xy+6y−3x= x3−8y3+27+18xy

(iv) . 9x2+25y2+15xy+12x−20y+16by3x−5y+4

SOLUTION  :

(3x)2+(5y)2+42−(−3x)(5y)−(5y)(4)−(4)(−3x)=(−3x)3+(5y)3+43−3(−3x)(5y)(4)

⇒9x2+25y2+16+15xy−20y+12x=−27x3+125y3+64+180xy

The document Ex-5.4, 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-5.4, Factorization Of Algebraic Expressions, Class 9, Maths RD Sharma Solutions - RD Sharma Solutions for Class 9 Mathematics

1. How do I factorize algebraic expressions in class 9?
Ans. To factorize algebraic expressions in class 9, you need to follow these steps: 1. Identify the common factors, if any, among the terms of the expression. 2. Take out the common factors from each term and write them outside the parentheses. 3. Divide each term by the common factors and write the remaining factors inside the parentheses. 4. Simplify the expression inside the parentheses, if possible, by further factorizing it. 5. Write the final factorized form of the expression.
2. What is the importance of factorization in algebraic expressions?
Ans. Factorization is important in algebraic expressions because it helps in simplifying and solving equations. It allows us to rewrite complex expressions in a simplified form, making it easier to perform mathematical operations. Factorization also helps in finding common factors, which can be useful in simplifying fractions and solving equations by canceling out common factors.
3. Can you give an example of factorizing an algebraic expression?
Ans. Sure! Let's factorize the expression 2x^2 + 4x: 1. Identify any common factors among the terms. In this case, both terms have a common factor of 2. 2. Take out the common factor from each term: 2(x^2 + 2x). 3. Divide each term by the common factor: 2(x(x + 2)). 4. Simplify the expression inside the parentheses, if possible. In this case, it is already simplified. 5. The final factorized form is 2(x(x + 2)).
4. Are there any specific methods or techniques for factorizing algebraic expressions?
Ans. Yes, there are specific methods and techniques for factorizing algebraic expressions. Some common techniques include: - Taking out the common factors - Grouping terms - Using special factorization formulas such as the difference of squares, perfect square trinomial, etc. - Applying the distributive property - Using trial and error method It is important to practice and understand these techniques to factorize different types of algebraic expressions effectively.
5. Can factorization be used to solve equations?
Ans. Yes, factorization can be used to solve equations. By factorizing an equation, we can rewrite it in a simplified form, which makes it easier to find the solutions. Once an equation is factorized, we can set each factor equal to zero and solve for the variables. The solutions obtained from this process are the solutions to the original equation. Factorization can be particularly useful in solving quadratic equations, where the zero product property can be applied to find the solutions.
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