## RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.5

**Other Exercises**

- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.1
- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.2
- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.3
- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4
- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.5
- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.6
- RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.7

**Multiply:**

**Question 1.
**

**(5x + 3) by (7x + 2)**

**Solution:**

(5x + 3) x (7x + 2)

= 5x (7x + 2) + 3 (7x + 2)

= 35x

^{2}+ 10x + 21x + 6

= 35x

^{2}+ 31x + 6

**Question 2.
**

**(2x + 8) by (x – 3)**

**Solution:**

(2x + 8) x (x – 3)

= 2x (x – 3) + 8 (x – 3)

= 2x

^{2}– 6x + 8x – 24

= 2x

^{2}+ 2x – 24

**Question 3.
**

**(7x +y) by (x + 5y)**

**Solution:**

(7x + y) x (x + 5y)

= 7x (x + 5y) + y (x + 5y)

= 7x

^{2}+ 35xy + xy + 5y

^{2 }=7x

^{2}+ 36xy + 5y

^{2}

**Question 4.
**

**(a – 1) by (0.1a**

^{2}+ 3)**Solution:**

(a – 1) x (0.1a

^{2}+ 3)

= a (0.1a

^{2}+ 3) – 1 (0.1a

^{2}+ 3)

= 0.1a

^{3}+ 3a-0.1a

^{2}-3

= 0.1a

^{3}– 0.1a

^{2}+ 3a-3

**Question 5.
**

**(3x**

^{2}+y^{2}) by (2x^{2}+ 3y^{2})**Solution:**

(3x

^{2}+y

^{2}) x (2x

^{2}+ 3y

^{2})

= 3x

^{2}(2x

^{2}+ 3y

^{2}) + y

^{2}(2x

^{2}+ 3y

^{2})

= 6x

^{2 +2}+ 9x

^{2}y

^{2}+ 2x

^{2}y

^{2}+ 3y

^{2}

^{+ 2 }= 6x

^{4}+ 11 x

^{2}y

^{2}+ 3y

^{4}

**Question 6.**

**Solution:**

**Question 7.
**

**(x**

^{6}-y^{6}) by (x^{2}+y^{2})**Solution:**

(x

^{6}– y

^{6}) x (x

^{2}+ y

^{2})

= x

^{6}(x

^{2}+ y

^{2}) – y

^{6}(x

^{2}+ y

^{2})

= x

^{6}x x

^{2}+ x

^{6}y

^{2}– x

^{2}y

^{6}-y

^{6}x y

^{2 }= x

^{6 + 2}+ x

^{6}y

^{2}– x

^{2}y

^{6}– y

^{6 }

^{+2 }= x + x

^{6}y

^{2}– x

^{2}y

^{6}– y

^{8}

**Question 8.
**

**(x**

^{2}+ y^{2}) by (3a+2b)**Solution:**

(x

^{2}+ y

^{2}) x (3a + 2b)

= x

^{2}(3a + 2b) + y

^{2}(3a + 2b)

= 3x

^{2}a + 2x

^{2}b + 3y

^{2}a + 2y

^{2}b

3ax

^{2}+ 3av

^{2}+ 2bx

^{2}+ 2by

^{2}

**Question 9.
**[-3d + (-7ƒ)] by (5d +ƒ)

**Solution:**

[-3d + (-7ƒ)] x (5d +ƒ)

= -3d x (5d +ƒ) + (-7ƒ) x (5d +ƒ)

= -15d

^{2}-3dƒ- 35dƒ- 7ƒ

^{2 }= -15d

^{2}– 38dƒ- 7ƒ

^{2}

**Question 10.
**

**(0.8a – 0.5b) by (1.5a -3b)**

**Solution:**

(0.8a – 0.5b) x (1.5a-3b)

= 0.8a x (1.5a – 36) – 0.56 (1.5a -3b)

= 1.2a

^{2}– 2.4ab – 0.75ab + 1.5b

^{2 }= 1.2a

^{2}-3.15ab+ 1.5b

^{2}

**Question 11.
**

**(2x**

^{2}y^{2}– 5xy^{2}) by (x^{2}-y^{2})**Solution:**

(2x

^{2}y

^{2}– 5xy

^{2}) x (x

^{2}-y

^{2})

= 2x

^{2}y

^{2}(x

^{2}– y

^{2}) – 5x_y

^{2}(x

^{2}– y

^{2})

= 2x

^{2}y

^{2}x x

^{2}– 2x

^{2}y

^{2}xy

^{2}– 5xy

^{2}x x

^{2}+ 5x

^{2}xy

^{2}

^{ }= 2x

^{2}

^{ + 2}y

^{2}– 2x

^{2}x y

^{2 + 2}– 5x

^{1+2 }y

^{2}+5xy

^{2}

^{ + 2 }= 2x

^{4}y

^{2}– 2x

^{2}y

^{4}– 5x

^{3}y

^{2}+ 5xy

^{4}

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.
**

**(2x**

^{2}-1) by (4x^{3}+ 5x^{2})**Solution:**

(2x

^{2}-1)x(4x

^{3}+ 5x

^{2})

= 2x

^{2}x (4x

^{3}+ 5x

^{2}) – 1 (4x

^{3}+ 5x

^{2})

= 2x

^{2}x 4x

^{3}+ 2x

^{2}x 5x

^{2}– 4x

^{3}– 5x

^{2 }= 8x

^{2 + 3}+ 10x

^{2 + 2}-4x

^{3}-5x

^{2 }= 8x

^{5}+ 10x

^{4}– 4x

^{3}– 5x

^{2}

**Question 16.
**

**(2xy + 3y**

^{2}) (3y^{2}– 2)**Solution:**

(2xy + 3y

^{2}) (3y

^{2}– 2)

= 2xy x (3y

^{2}-2) + 3y

^{2}x (3y

^{2}-2)

= 2xy x Zy

^{2}+ 2xy x (-2) + Zy

^{2}x Zy

^{2}– Zy

^{2}x 2

= 6xy

^{1 + 2}– 4xy + 9y

^{2 + 2}– 6y

^{2 }= 6xy

^{3}– 4xy + 9y

^{4}– 6y

^{2 }Find the following products and verify the result for x = -1, y = -2 :

**Question 17.
**

**(3x-5y)(x+y)**

**Solution:**

(3x-5y)(x+y)

= 3x x (x + y) – 5y x (x + y)

= 3x x x + 3x x y-5y x x-5y x y

= 3x

^{2}+ 3xy – 5xy – 5y

^{2 }= 3x

^{2}– 2xy – 5y

^{2 }Verfification:

x = -1,y = -2

L.H.S. = (3x-5y)(x+y)

= [3 (-1) -5 (-2)] [-1 – 2]

= (-3 + 10) (-3) = 7 x (-3) = -21

R.H.S. = 3x

^{2}– 2xy – 5y

^{2}

= 3 (-1)

^{2}– 2 (-1) (-2) -5 (-2)

^{2 }=3×1-4-5×4=3-4-20

= 3-24 = -21

∴ L.H.S. = R.H.S.

**Question 18.
**

**(x**

^{2}y-1) (3-2x^{2}y)**Solution:**

(x

^{2}y-1) (3-2x

^{2}y)

= x

^{2}y (3 – 2x

^{2}y) -1(3-2x

^{2}y)

= x

^{2}y x 3 – x

^{2}y x 2x

^{2}y – 1 x 3 + 1 x 2x

^{2}y

= 3x

^{2}y-2x

^{2 + 2}x y

^{1}

^{ +1}-3 + 2x

^{2}y

= 3x

^{2}y – 2x

^{4}y

^{2}– 3 + 2x

^{2}y

= 3x

^{2}y + 2x

^{2}y – 2x

^{4}y

^{2}– 3

= 5x

^{2}y – 2x

^{4}y

^{2}– 3

Verification : (x = -1, y = -2)

L.H.S. = (x

^{2}y – 1) (3 – 2x

^{2}y)

= [(-1)

^{2}x (-2) -1] [3 – 2 x (-1)2 x (-2)]

= [1 x (-2) -1) [3 – 2 x 1 x (-2)]

= (-2 – 1) (3 + 4) = -3 x 7 = -21

R.H.S. = 5x

^{2}y – 2x

^{4}y

^{2}– 3

= 5 (-1)

^{2}(-2) -2 (-1)

^{4}(-2)

^{2}-3

5 x 1 (-2) – 2 (1 x 4) -3

= -10-8-3 = -21

∴ L.H.S. = R.H.S

**Question 19.**

**Solution:**

**Simplify :**

**Question 20.
**

**x**

^{2}(x + 2y) (x – 3y)**Solution:**

x

^{2}(x + 2y) (x – 3y)

= x

^{2}[x (x – 3y) + 2y (x – 3y)]

= x

^{2}[x

^{2}– 3xy + 2xy – 6y

^{2}]

= x

^{2}[x

^{2}– xy – 6y

^{2})

= x

^{2}x x

^{2}– x

^{2}x xy – x

^{2}6y

^{2 }= x

^{4}– x

^{3}y – 6x

^{2}y

^{2}

**Question 21.
**

**(x**

^{2}– 2y^{2}) (x + 4y)**Solution:**

(x

^{2}– 2y

^{2}) (x + 4y) x

^{2}y

^{2 }= [x

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

^{2}(x + 4y)] x

^{2}y

^{2 }= (x

^{3}+ 4x

^{2}y – 2xy

^{2}– 8y

^{3}) x

^{2}y

^{2 }= x

^{2}y

^{2}x x

^{3}+ x

^{2}y

^{2}x 4x

^{2}y – 2x

^{2}y

^{2}x xy

^{2}– 8x

^{2}y

^{2}x y

^{3 }= x

^{2 +3}y

^{2}+ 4x

^{2 + 2 }y

^{2 +1}– 2x

^{2 +1}y

^{2+ 2}– 8x

^{2}y

^{2+}

^{3 }= xy + 4

^{4}xy

^{3}– 2x

^{3}y

^{4}– 8x

^{2}y

^{5}

**Question 22.
**

**a**

^{2}b^{2}(a + 2b) (3a + b)**Solution:**

a

^{2}b

^{2}(a + 2b) (3a + b)

= a

^{2}b

^{2}[a (3a + b).+ 2b (3a + b)]

= a

^{2}b

^{2}[3a

^{2}+ ab + 6ab + 2b

^{2}]

= a

^{2}b

^{2}[3a

^{2}+ lab + 2b

^{2}]

= a

^{2}b

^{2}x 3a

^{2}+ a

^{2}b

^{2}x 7ab + a

^{2}b

^{2}x 2b

^{2 }= 3a

^{2 + }

^{2}b

^{2}+ 7a

^{2+1 }b

^{2+1}+ 2a

^{2}b

^{2}

^{ + 2 }= 3a

^{4}b

^{2}+ 7a

^{3}b

^{3}+ 2a

^{2}b

^{4}

**Question 23.
**

**x**

^{2}(x-y) y^{2}(x + 2y)**Solution:**

x

^{2}(x -y) y

^{2}(x + 2y)

= [x

^{2}x x – x

^{2}x y] [y

^{2}x x + y

^{2}x 2y]

= (x

^{3}– x

^{2}y) (xy

^{2}+ 2y

^{3})

= x

^{3}(xy

^{2}+ 2y

^{3}) – x

^{2}y (xy

^{2}+ 2y

^{3})

= x

^{3}x xy

^{2}+ x

^{3}x 2y

^{3}– x

^{2}y x xy

^{2}– x

^{2}y x 2y

^{3}

= x

^{3 +1 }y

^{2}+ 2x

^{3}y

^{3}– x

^{2 +1}y

^{1+ 2}– 2x

^{2}y

^{1 + 3 }= x

^{4}y

^{2}+ 2x

^{3}y

^{3}– x

^{3}y

^{3}– 2x

^{2}y

^{4}

= x

^{4}y

^{2}+ x

^{3}y

^{3}– 2x

^{2}y

^{4}

**Question 24.
**

**(x**

^{3}– 2x^{2}+ 5x-7) (2x-3)**Solution:**

(x

^{3}– 2x

^{2}+ 5x – 7) (2x – 3)

= (2x – 3) (x

^{3}– 2x

^{2}+ 5x – 7)

= 2x (x

^{3}– 2x

^{2}+ 5x – 7) -3 (x

^{3}– 2x

^{2}+ 5x – 7)

= 2x x x

^{3}– 2x x 2x

^{2}+ 2x x 5x – 2x x 7 -3 x x

^{3}– 3 x (-2x

^{2}) – 3 x 5x – 3 x (-7)

= 2x

^{4}-4x

^{3}+ 10x

^{2}– 14x-3x

^{3}+ 6x

^{2}– 15x + 21

= 2x

^{4}– 4x

^{3}– 3x

^{3}+ 10x

^{2}+ 6x

^{2}– 14x- 15x + 21

= 2x

^{4}-7x

^{3}+ 16x

^{2}-29x+ 21

**Question 25.
**

**(5x + 3) (x – 1) (3x – 2)**

**Solution:**

(5x + 3) (x – 1) (3x – 2)

= (5x + 3) [x (3x – 2) -1 (3x – 2)]

= (5x + 3) [3x

^{2}– 2x – 3x + 2]

= (5x + 3) [3x

^{2}– 5x + 2]

= 5x (3x

^{2}– 5x + 2) + 3 (3x

^{2}– 5x + 2)

= (5x x 3x

^{2}– 5x x Sx + 5x x 2)+ [3 x 3x

^{2}+ 3 x (-5x) + 3×2]

= 15x

^{3}– 25x

^{2}+ 10x + 9x

^{2}– 15x + 6

= 15x

^{3}– 25x

^{2}+ 9x

^{2}+ 10x – 15x + 6

= 15x

^{3}– 16x

^{2}– 5x + 6

**Question 26.
**

**(5-x) (6-5x) (2-x)**

**Solution:**

(5-x) (6-5x) (2-x)

= [5 (6 – 5x) -x (6 – 5x)] (2 – x)

= [30 – 2$x – 6x + 5x

^{2}] (2 – x)

= (30 – 3 1x + 5x

^{2}) (2-x)

= 2 (30 – 31x + 5x

^{2}) – x (30 – 31x + 5x

^{2})

= 60 – 62x + 10x

^{2}– 30x + 3 1x

^{2}– 5x

^{3 }= 60 – 62x – 30x + 10x

^{2}+ 3 1x

^{2}– 5x

^{3 }= 60 – 92x + 41x

^{2}– 5x

^{3}

**Question 27.
**

**(2x**

^{2}+ 3x – 5) (3x^{2}– 5x + 4)**Solution:**

(2x

^{2}+ 3x – 5) (3x

^{2}– 5x + 4)

= 2x

^{2}(3x

^{2}– 5x + 4) + 3x (3x

^{2}– 5x + 4) -5 (3x

^{2}– 5x + 4)

= 2x

^{2}x 3x

^{2}– 2x

^{2}x 5x + 2x

^{2}x 4 + 3x x 3x

^{2}– 3x x 5x + 3x x 4 – 5 x 3x

^{2}– 5 (-5x) -5×4

= 6x

^{4}– 10x

^{3}+ 8x

^{2}+ 9x

^{3}– 15x

^{2}+ 12x – 15x

^{2 }+ 25x-20

= 6x

^{4}– 10x

^{3}+ 9x

^{3}+ 8x

^{2}– 15x

^{2}– 15x

^{2}+ 12x + 25x – 20

= 6x

^{4}– x

^{3}– 22x

^{2}+ 37x – 20

**Question 28.
**

**(3x – 2) (2x – 3) + (5x – 3) (x + 1)**

**Solution:**

(3x – 2) (2x – 3) + (5x – 3) (x + 1)

= 3x (2x – 3) -2 (2x – 3) + 5x (x + 1) – 3 (x + 1)

= 6x

^{2}– 9x – 4x + 6 + 5x

^{2}+ 5x – 3x – 3

= 6x

^{2}+ 5x

^{2}– 9x – 4x + 5x – 3x + 6 – 3

= 11x

^{2}– 11x + 3

**Question 29.
**

**(5x – 3) (x + 2) – (2x + 5) (4x – 3)**

**Solution:**

(5x – 3) (x + 2) – (2x + 5) (4x – 3)

= [5x (x + 2) -3 (x + 2)] – [2x (4x – 3) + 5 (4x – 3)]

= [5x

^{2}+ 1 0x – 3x – 6] – [8x

^{2}– 6x + 20x -15]

= (5x

^{2}+ 7x – 6) – (8x

^{2}+ 14x – 15)

= 5x

^{2}+ lx – 6 – 8x

^{2}– 14x + 15

= 5x

^{2}– 8x

^{2}+ 7x – 14x – 6 + 15

= -3x

^{2}– 7x + 9

**Question 30.
**

**(3x + 2y) (4x + 3y) – (2x – y) (7x – 3y)**

**Solution:**

(3x + 2y) (4x + 3y) – (2x – y) (7x – 3y)

= [3x (4x + 3y) + 2y (4x + 3y)]-[2x (7

_{x}-3y)-y(7

_{x}-3y)]

= (12x

^{2}+ 9xy + 8xy + 6y

^{2}) – (14x

^{2}– 6xy – 7xy + 3y

^{2})

= (12x

^{2}+ 17xy + 6y

^{2}) – (14x

^{2}– 13xy + 3y

^{2})

= 12x

^{2}+ 17xy + 6y

^{2}– 14x

^{2}+ 13xy – 3y

^{2 }= 12x

^{2}– 14x

^{2}+ 17xy + 13xy + 6y

^{2}– 3y

^{2 }= -2x

^{2}+ 30xy + 3y

^{2}

= -2x

^{2}+ 3y

^{2 }+ 30xy

**Question 31.
**

**(x**

^{2}-3x + 2) (5x- 2) – (3x^{2}+ 4x-5) (2x- 1)**Solution:**

(x

^{2}-3x + 2) (5x- 2) – (3x

^{2}+ 4x-5) (2x- 1)

= [5x (x

^{2}– 3x + 2) -2 (x

^{2}– 3x + 2)] – [2x (3x

^{2}+ 4x – 5) -1 (3x

^{2}+ 4x – 5)]

= [5x

^{3}– 15x

^{2}+ 10x – 2x

^{2}+ 6x – 4] – [6x

^{3}+ 8x

^{2}– 10x – 3x

^{2}– 4x + 5]

= [5x

^{3}– 15x

^{2}– 2x

^{2}+ 10xc + 6x – 4] – [6x

^{3}+ 8x

^{2}– 3x

^{2}– 10x – 4x + 5]

= (5x

^{3}– 17x

^{2}+ 16x-4) – (6x

^{3}+ 5x

^{2}– 14x + 5)

= 5x

^{3}– 17x

^{2}+ 16x – 4 – 6x

^{3}– 5x

^{2}+ 14x – 5

= 5x

^{3}– 6x

^{3}– 17x

^{2}– 5x

^{2}+ 16x + 14x – 4 – 5

= -x

^{3}– 22x

^{2}+ 30x – 9

**Question 32****.
**

**x**

^{3}– 2x^{2}+ 3x – 4) (x – 1) – (2x – 3) (x^{2}– x + 1)**Solution:**

(x

^{3}– 2x

^{2}+ 3x – 4) (x – 1) – (2x – 3) (x

^{2}– x + 1)

= [x (x

^{3}– 2x

^{2}+ 3x – 4) – 1 (x

^{3}– 2x

^{2}+ 3x – 4)] – [2x (x

^{2}– x + 1) – 3 (x

^{2}– x + 1)]

= [x

^{4}– 2x

^{3}+ 3x

^{2}– 4x – x

^{3}+ 2x

^{2}– 3x + 4] [2x

^{3}– 2x

^{2}+ 2x – 3x

^{2}+ 3x – 3]

= (x

^{4}– 2x

^{3}– x

^{3}+ 3x

^{2}+ 2x

^{2}– 4x – 3x + 4) (2x

^{3}– 2x

^{2}– 3x

^{2}+ 2x + 3x – 3)

= (x

^{4}– 3x

^{3}+ 5x

^{2}– 7x + 4) – (2x

^{3}– 5x

^{2}+ 5x – 3)

= x

^{4}– 3x

^{3}+ 5x

^{2}– 7x + 4 – 2x

^{3}+ 5x

^{2}– 5x + 3

= x

^{4}– 3x

^{3}– 2x

^{3}+ 5x

^{2}+ 5x

^{2}– 7x – 5x + 4 + 3

= x

^{4}– 5x

^{3}+ 10x

^{2}– 12x + 7

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