## RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.1

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.1

Other Exercises

- RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.1
- RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.2
- RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.3
- RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.4
- RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions VSAQS
- RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions MCQS

Factorize

Question 1.

x^{3} + x – 3x^{2} – 3

Solution:

x^{3} + x – 3x^{2} – 3

x^{3} – 3a^{2} + x – 3

⇒ x^{2}(x – 3) + 1(x – 3)

= (x – 3) (x^{2} + 1)

Question 2.

a(a + b)^{3} – 3a^{2}b(a + b)

Solution:

a(a + b)^{3} – 3a^{2}b(a + b)

= a(a + b) {(a + b)^{2} – 3ab}

= a(a + b) {a^{2} + b^{2} + 2ab – 3ab}

= a{a + b) {a^{2} – ab + b^{2})

Question 3.

x(x^{3} – y^{3}) + 3xy(x – y)

Solution:

x(x^{3} – y^{3}) + 3xy(x – y)

= x(x – y) (x^{2} + xy + y^{2}) + 3xy(x – y)

= x(x – y) (x^{2} + xy + y^{2} + 3y)

= x(x – y) (x^{2} + xy + y^{2} + 3y)

Question 4.

a^{2}x^{2} + (ax^{2} +1)x + a

Solution:

a^{2}x^{2} + (ax^{2} + 1)x + a

= a^{2}x^{2} + a + (ax^{2} + 1)x

= a(ax^{2} + 1) + x(ax^{2} + 1)

= (ax^{2} + 1) (a + x)

= (x + a) (ax^{2} + 1)

Question 5.

x^{2} + y – xy – x

Solution:

x^{2} + y – xy – x

= x^{2}-x-xy + y = x(x- l)-y(*- 1)

= (x – 1) (x – y)

Question 6.

X^{3} – 2x^{2}y + 3xy^{2} – 6y^{3
}Solution:

x^{3} – 2x^{2}y + 3xy^{2} – 6y^{3}

= x^{2}(x – 2y) + 3y^{2}(x – 2y)

= (x – 2y) (x^{2} + 3y^{2})

Question 7.

*6*ab – b^{2} + 12ac – 2bc

Solution:

6ab – b^{2} + 12ac – 2bc

= 6ab + 12ac – b^{2} – 2bc

= 6a(b + 2c) – b(b + 2c)

= (b + 2c) (6a – b)

Question 8.

x(x – 2) (x – 4) + 4x – 8

Solution:

x(x – 2) (x – 4) + 4x – 8

= x(x – 2) (x – 4) + 4(x – 2)

= (x – 2) [x(x – 4) + 4]

= (x – 2) (x^{2} – 4x + 4)

= (x – 2) [(x)^{2} – 2 x x x 2 + (2)^{2}]

= (x – 2) (x – 2)^{2} = (x – 2)^{3}

Question 9.

(a – b + c)^{2} + (b – c + a)^{2} + 2(a – b + c) (b – c + a)

Solution:

(a – b + c)^{2} + ( b- c+a)^{2} + 2(a – b + c) (b – c + a) {∵ a^{2} + b^{2} + 2ab = (a + b)^{2}}

= [a – b + c + b- c + a]^{2
}= (2a)^{2} = 4a^{2}

Question 10.

a^{2} + 2ab + b^{2} – c^{2 }

Solution:

a^{2} + 2ab + b^{2} – c^{2
}= (a^{2} + 2ab + b^{2}) – c^{2
}= (a + b)^{2} – (c)^{2 }{∵ a^{2} – b^{2} = (a + b) (a – b)}

= (a + b + c) (a + b – c)

Question 11.

a^{2} + 4b^{2} – 4ab – 4c^{2 }

Solution:

Question 12.

x^{2} – y^{2} – 4xz + 4z^{2
}Solution:

x^{2} – y^{2} – 4xz + 4z^{2
}= x^{2} – 4xz + 4z^{2} – y^{2
}= (x)^{2} – 2 x x x 2z + (2z)^{2} – (y)^{2
}= (x – 2z)^{2} – (y)^{2
}= (x – 2z + y) (x – 2z – y)

= (x +y – 2z) (x – y – 2z)

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Give possible expression for the length and breadth of the rectangle having 35y^{2} + 13y – 12 as its area.

Solution:

Area of a rectangle = 35y^{2} + 13y – 12

= 35y^{2} + 28y- 15y- 12

(i) If length = 5y + 4, then breadth = 7y – 3

(ii) and if length = 7y-3, then length = 5y+ 4

Question 17.

What are the possible expressions for the dimensions of the cuboid whose volume is 3x^{2} – 12x.

Solution:

Volume 3x^{2} – 12x

= 3x(x – 4)

∴ Factors are 3, x, and x – 4

Now, if length = 3, breadth = x and height = x – 4

if length =3, breadth = x – 4, height = x

if length = x, breadth = 3, height = x – 4

if length = x, breadth = x – 4, height = 3

if length = x – 4, breadth = 3, height = x

if length – x – 4, breadth = x, height = 3

Question 18.

Solution:

Question 19.

(x + 2) (x^{2} + 25) – 10x^{2} – 20x

Solution:

(x + 2) (x^{2} + 25) – 10x^{2} – 20x

= (x + 2) (x^{2} + 25) – 10x(x + 2)

= (x + 2) [x^{2} + 25 – 10x]

= (x + 2) [(x)^{2} – 2 x x x 5 + (5)^{2}]

= (x + 2) (x – 5)^{2}

Question 20.

2a^{2} + 2\(\sqrt { 6 } \) ab +3b^{2
}Solution:

2a^{2} + 2\(\sqrt { 6 } \) ab +3 b^{2
}= (\(\sqrt { 2 } \) a)^{2}+ \(\sqrt { 2 } \) a x \(\sqrt { 3 } \) b+ (\(\sqrt { 3 } \) b)^{2
}= (\(\sqrt { 2 } \)a + \(\sqrt { 3 } \) b)2

Question 21.

a^{2} + b^{2} + 2(ab + bc + ca)

Solution:

a^{2} + b^{2} + 2(ab + bc + ca)

= a^{2} + b^{2} + 2 ab + 2 bc + 2 ca

= (a + b)^{2} + 2c(b + a)

= (a + b)^{2} + 2c(a + b)

= (a + b) (a + b + 2c)

Question 22.

4(x – y)^{2} – 12(x -y) (x + y) + 9(x + y)^{2
}Solution:

4(x – y)^{2} – 12(x – y) (x + y) + 9(x + y)^{2
}= [2(x – y)^{2} + 2 x 2(x – y) x 3(x + y) + [3 (x+y]^{2 }{∵ a^{2} + b^{2} + 2 abc = (a + b)^{2}}

= [2(x – y) + 3(x + y)]^{2
}= (2x-2y + 3x + 3y)^{2}

= (5x + y)^{2}

Question 23.

a^{2} – b^{2} + 2bc – c^{2}

Solution:

a^{2} – b^{2} + 2bc – c^{2
}= a^{2} – (b^{2} – 2bc + c^{2}) {∵ a^{2} + b^{2} – 2abc = (a – b)^{2}}

= a^{2} – (b – c)^{2
}= (a)^{2} – (b – c)^{2 }{∵ a^{2} – b^{2} = (a + b) (a – b)}

= (a + b – c) (a – b + c)

Question 24.

xy^{9} – yx^{9
}Solution:

xy^{9} – yx^{9} = xy(y^{8} – x^{8})

= -xy(x^{8} – y^{8})

= -xy[(x^{4})^{2} – (y^{4})^{2}]

= -xy (x^{4} + y^{4}) (x^{4} – y^{4}) {∵ a^{2}-b^{2} = (a + b) (a – b)}

= -xy (x^{4} + y^{4}) {(x^{2})^{2} – (y^{2})^{2}}

= -xy(x^{4 }+ y^{4}) (x^{2} + y^{2}) (x^{2} – y^{2})

= -xy (x^{4} +y^{4}) (x^{2} + y^{2}) (x + y) (x -y)

= -xy(x – y) (x + y) (x^{2} + y^{2}) (x^{4} + y^{4})

Question 25.

x^{4} + x^{2}y^{2} + y^{4
}Solution:

x^{4} + x^{2}y^{2} + y^{4} = (x^{2})^{2} + 2x^{2}y^{2} + y^{4} – x^{2}y^{2 }(Adding and subtracting x^{2}y^{2})

= (x^{2} + y^{2})^{2} – (xy)^{2 }{∵ a^{2} – b^{2} = (a + b) (a – b)}

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

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

Question 26.

x^{2} + 6\(\sqrt { 2 } \)x + 10

Solution:

Question 27.

x^{2} + 2\(\sqrt { 2 } \)x- 30

Solution:

Question 28.

x^{2} – \(\sqrt { 3 } \)x – 6

Solution:

Question 29.

x^{2} + 5 \(\sqrt { 5 } \)x + 30

Solution:

Question 30.

x^{2} + 2 \(\sqrt { 3 } \)x – 24

Solution:

Question 31.

5 \(\sqrt { 5 } \)x^{2} + 20x + 3\(\sqrt { 5 } \)

Solution:

Question 32.

2x^{2} + 3\(\sqrt { 5 } \) x + 5

Solution:

Question 33.

9(2a – b)^{2} – 4(2a – b) – 13

Solution:

Question 34.

7(x-2y) – 25(x-2y) +12

Solution:

Question 35.

2(x+y) – 9(x+y) -5

Solution:

2(x+y) – 9(x+y) -5

Hope given RD Sharma Class 9 Solutions Chapter 5 Factorisation of Algebraic Expressions Ex 5.1 are helpful to complete your math homework.

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