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Textbook | NCERT |

Class | Class 8 |

Subject | Maths |

Chapter | Chapter 8 |

Chapter Name | Division of Algebraic Expressions |

Exercise | Ex 8.5 |

**RD Sharma Solutions for Class 8 Chapter 8 Division of Algebraic Expressions Ex 8.5** **Download PDF**

**Question 1.**

**Divide the first polynomial by the second polynomial in each of the following. Also, write the quotient and remainder.**

**(i) 3x**

^{2}+ 4x + 5, x – 2**(ii) 10x**

^{2}– 7x + 8, 5x – 3**(iii) 5y**

^{3}– 6y^{2}+ 6y-1,5y-1**(iv)x**

^{4}-x^{3}+ 5x,x-1**(v) y**

^{4}+y^{2},y^{2}-2**Solution:**

**(i)**3x

^{2}+ 4x + 5, x – 2

= 3x (x – 2) + 10x + 5

= 3x (x – 2) + 10 (x – 2) + 25

∴ Quotient = 3x + 10

Remainder = 25

**(iii)**5y

^{3}– 6y

^{2}+ 6y – 1, 5y – 1

= y

^{2 }(5y – 1) – 5y

^{2}+ 6y- 1

= y

^{2}(5y – 1) -y (5y – 1) + 5y – 1

= y

^{2}(5y- 1) -y (5y- 1) + 1 (5y- 1)

∴ Quotient = y

^{2}– y + 1 and Remainder = 0

**(iv)**x

^{4}– x

^{3}+ 5x, x – 1

= x

^{3}(x – 1) + 5x

= x

^{3}(x – 1) + 5 (x – 1) + 5

∴ Quotient = x

^{3}+ 5, Remainder = 5

**(v)**y

^{4}+y

^{2},y

^{2}– 2

= y

^{2}(y

^{2 }– 2) + 3y

^{2}= y

^{2}(y

^{2}– 2) + 3 (y

^{2}– 2) + 6

∴ Quotient =y

^{2}+ 3 and Remainder = 6

**Question 2.Find, whether or not the first polynomial is a factor of the second :**

**(i) x + 1, 2x**

^{2}+ 5x + 4**(ii) y- 2, 3y**

^{3}+ 5y^{2}+ 5y + 2**(iii) 4x**

^{2}– 5, 4.x^{4}+ 7x^{2}+ 15**(iv) 4-z, 3z**

^{2}– 13z + 4**(v) 2a-3,10a**

^{2}– 9a – 5**(vi) 4y+1 ,8y**

^{2}-2y + 1**Solution:**

**(i)**x + 1, 2x

^{2}+ 5x + 4

2x

^{2}+ 5x + 4 = 2x (x + 1) + 3x + 4

= 2x (x + 1) + 3 (x + 1) + 1

∵ Remainder = 1

∴ x + 1 is not a factor of 2x

^{2}+ 5x + 4

**(ii)**y – 2, 3y

^{3}+ 5y

^{2}+ 5y + 2

3y

^{3}+ 5y

^{2}+ 5y + 2 = 3y

^{2}(y – 2)+11y

^{2}+ 5y + 2

= 3y

^{2}(y – 2)+11y (y – 2) + 27y + 2

= 3y

^{2}(y – 2) + 11y (y – 2) + 27 (y – 2) + 56

∵ Remainder = 56

∴ y – 2 is not a factor of 3y

^{3}+ 5y

^{2}+ 5y + 2

**(iii)**4x

^{2}– 5, 4x

^{4}+ 7x

^{2}+ 15

4x

^{4}+ 7x

^{2}+ 15 = x

^{2}(4x

^{2}– 5) + 12x

^{2}+ 15

= x

^{2}(4x

^{2}– 5) + 3 (4x

^{2}– 5) + 30

∵ Remainder = 30

∴ 4x

^{2}– 5 is not a factor of 4x

^{4}+ 7x

^{2}+ 15

**(iv)**4 – z, 3z

^{2}– 13z + 4

3z

^{2}– 13z + 4 = -3z (-z + 4) – z + 4

= -3z (-z + 4) + 1 (-z + 4)

∵ Remainder = 0

∴ 4 – z or – z + 4 is a factor of 3z

^{2}– 13z + 4

**(v)**2a – 3, 10a

^{2}– 9a – 5

10a

^{2}– 9a – 5 = 5a (2a – 3) + 6a – 5

= 5a (2a – 3) + 3 (2a – 3) + 4

∵ Remainder = 4

∴ 2a – 3 is not a factor of 10a

^{2}– 9a – 5

**(vi)**4y + 1, 8y

^{2}– 2y + 1

8y

^{2}– 2y + 1 = 2y (4y + 1) – 4y + 1

= 2y (4y + 1) – 1 (4y + 1) + 2

∵ Remainder = 2

∴ 4y + 1 is not a factor of 8y

^{2}– 2y + 1

**All Chapter RD Sharma Solutions For Class 8 Maths**

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