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Division Of Polynomials

Class 8th Mathematics (old) MHB Solution
Exercise 80
  1. Complete the following statements by filling in the blanks.(1) 21 ÷ 7 … ∵ 21 = 7 × 3(2) 45…
Exercise 81
  1. 21b; 4 Divide the first polynomial by the second. Explain the steps.…
  2. 36p^2 ; 6p Divide the first polynomial by the second. Explain the steps.…
  3. 5a^3 ; a^3 Divide the first polynomial by the second. Explain the steps.…
  4. -30n^2 ; 3n Divide the first polynomial by the second. Explain the steps.…
  5. 15m^3 ; 5m Divide the first polynomial by the second. Explain the steps.…
  6. 2b^5 ; b^2 Divide the first polynomial by the second. Explain the steps.…
  7. 78x^4 ; -7x^2 Divide the first polynomial by the second. Explain the steps.…
  8. -8y^5 ; -y^3 Divide the first polynomial by the second. Explain the steps.…
  9. 4n^4 ÷ n^2 Divide.
  10. 52m^3 ÷ 13m Divide.
  11. 20c^2 ÷ 15c Divide.
  12. 36x^5 ÷ 4x^3 Divide.
  13. 40a^2 ÷ (-10a) Divide.
  14. (-33y^4) ÷ (-11y) Divide.
  15. Write five different pairs of dividends and divisors such that the quotient in…
Exercise 82
  1. Fill in the blanks in the division (- 7b^3 + 3b^5) / 3b^3 shown below. 3b^5 =…
  2. n^2 + 3 n Divide the binomial by the monomial. Write the quotient and the…
  3. p^2 - 6 p Divide the binomial by the monomial. Write the quotient and the…
  4. 3x^2 - 15x 3x Divide the binomial by the monomial. Write the quotient and the…
  5. 2y^3 + 5y^2y^2 Divide the binomial by the monomial. Write the quotient and the…
  6. 8a^3 - 12a^2 2a^2 Divide the binomial by the monomial. Write the quotient and…
  7. 18b^4 + 13b^3 2b^2 Divide the binomial by the monomial. Write the quotient and…
  8. 24m^4 - 16m^2 8m Divide the binomial by the monomial. Write the quotient and…
  9. 24m^4 - 16m^2 8m^2 Divide the binomial by the monomial. Write the quotient and…
  10. (12p^3 - 8p) / 4p Divide.
  11. (9c^2 - 21c) / 3c Divide.
  12. (7x^3 + 28x) / 7x Divide.
  13. (y^4 - 5y^2) / y^2 Divide.
  14. (1b^4 - 7b^2) / 3b^2 Divide.
  15. (3a^5 + a^3) / a^3 Divide.
  16. (18p^7 + 15p^5) / 3p^4 Divide.
  17. (8m^6 - 20m^4) / 8m^3 Divide.
  18. (8n^3 + 10n^2) / n^2 Divide. Write the answer in the form ‘Dividend = divisor…
  19. Divide. Write the answer in the form ‘Dividend = divisor × quotient +…
  20. (3p^3 + 6p^2) / 3p^2 Divide. Write the answer in the form ‘Dividend = divisor…
  21. (5x^4 + 5x^2) / 5x^2 Divide. Write the answer in the form ‘Dividend = divisor…
  22. (11y^4 - 4y) / y^2 Divide. Write the answer in the form ‘Dividend = divisor ×…
  23. (13a^4 - 26a^2) / 13a^2 Divide. Write the answer in the form ‘Dividend =…
  24. (15y^8 + 10y^6) / y^6 Divide. Write the answer in the form ‘Dividend = divisor…
  25. (27m^5 - 16m^2) / 3m Divide. Write the answer in the form ‘Dividend = divisor…
Exercise 83
  1. (m^3 + m^2 + m) / m Divide. Write the quotient and remainder.
  2. (2n^2 - 12n+8) / 2n Divide. Write the quotient and remainder.
  3. (10-25a+5a^2) / 5a Divide. Write the quotient and remainder.
  4. (13-8b^2 + 12b^3) / 4b Divide. Write the quotient and remainder.
  5. (p^4 - 2p+9) / p^2 Divide. Write the quotient and remainder.
  6. (7x^4 - 21x^2 + 6) / 7x^2 Divide. Write the quotient and remainder.…
  7. 6y^2 - 18y+7 3y Divide the trinomial by the monomial. Write the answer in the…
  8. (x^4 - x^3 + x^2) x^2 Divide the trinomial by the monomial. Write the answer…
Exercise 84
  1. x^2 - 7x+12 x-3 Divide the polynomial by the binomial. Write your answer in…
  2. 2y^3 + 6y^2 + 12y+8 y+1 Divide the polynomial by the binomial. Write your…
  3. 5a-12+3a^2 - 5a^3 + 6a^4a^2 - 1 Divide the polynomial by the binomial. Write…
  4. b^4 - 12b^2 :b^2 + 1 Divide the polynomial by the binomial. Write your answer…
  5. 12m^2 - m-17 3m+2 Divide the polynomial by the binomial. Write your answer in…
  6. 411-4n^2 + n^3 - n^4n^3 - 4 Divide the polynomial by the binomial. Write your…
  7. x^3 - 6x^2 + 11x-6 x-3 Divide the polynomial by the binomial. Write your…
  8. a^3 + 8 a+2 Divide the polynomial by the binomial. Write your answer in the…
  9. Divide the polynomial x^4 - 5x^3 + 5x^2 + 5x-6 by each of the following…
  10. Divide the polynomial x^4 - 5x^3 + 5x^2 + 5x-6 by each of the following…
  11. Divide the polynomial x^4 - 5x^3 + 5x^2 + 5x-6 by each of the following…
  12. Divide the polynomial x^4 - 5x^3 + 5x^2 + 5x-6 by each of the following…
  13. Divide the polynomial x^4 - 5x^3 + 5x^2 + 5x-6 by each of the following…
Exercise 80
Question 1.

Complete the following statements by filling in the blanks.

(1) 21 ÷ 7 … ∵ 21 = 7 × 3

(2) 45 ÷ 15 = …. ∵ 45 …… × ……

(3) We continue to carry out division until we get remainder zero or a whole number less than the ………….

(4) Dividend = division × (….) + (….), 0 ≤ remainder < divisor

(5) When we divide 43 by 5, the quotient is ………… and the remainder is ……….

Here, dividend = ………., divisor = ………

∴ 43 = 5 × …… + ……

(6) 2y × ……… = 10y2

(7) 5x × ……… = 15x4

(8) m2 × ……… = 9m4

(9) 3m × ………… = 9m4+
Answer:

(1) Here,


21/7


We know that 7 when multiplied by 3, we get 21 as the result.


So when 21 is divided by 7 we get 3 as the quotient.


So the answer is


21÷7 = 3 Q 21 = 7 × 3


(2) Here,


45/15


We know that 15 when multiplied by 3, we get 45 as the result.


So when 45 is divided by 15 we get 3 as the quotient.


So the answer is


45÷15 = 3 Q 45 = 15 × 3


(3) We continue to carry out division until we get remainder zero or a whole number less than the divisor.


(4) The above relation is as follows:


Dividend = Divisor × Quotient + Remainder


The number which we divide is called the dividend and the number by which we divide is called as divisor.


The result obtained is called as quotient and the number leftover is called as the remainder.


(5) For any number to be perfectly divisible by 5, the number should end by 5 or zero.


So 43 is not perfectly divisible by 5.


When we divide 43 by 5, the quotient is 8 and the remainder is 3.



When we divide 43 by 5, the quotient is 8 and the remainder is 3


Here, dividend = 43, divisor = 5


43 = 5 × 8 + 3


(6) For the polynomial 2y,


When 2y is multiplied 5y,


2× 5 = 10 and y × y = y2


So,


2y × 5y = 10 y2


(7) For the polynomial 5x,


When 5x is multiplied 3x3,


3 × 5 = 15 and x3 × x = x4


So,


5x × 3x3 = 15 x4


(8) For the polynomial m2,


When m2 is multiplied 9 m2,


1 × 9 = 9 and m2 × m2 = m4


So,


m2 × 9 m2 = 9 m4


(9) For the polynomial 3m,


When 3m is multiplied 3 m3,


3 × 3 = 9 and m3 × m = m4


So,


3m × 3 m3 = 9 m4
Exercise 81
Question 1.

Divide the first polynomial by the second. Explain the steps.

21b; 4
Answer:

The division is as follows:



Here when 7 is multiplied by 3b we get 21b as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 21b becomes perfectly divisible by 7 and hence the quotient is 3b and remainder is 0.
Question 2.

Divide the first polynomial by the second. Explain the steps.

36p2; 6p
Answer:

The division is as follows:



Here when 6p is multiplied by 6p we get 36p2 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 36p2 becomes perfectly divisible by 6p and hence the quotient is 6p and remainder is 0.
Question 3.

Divide the first polynomial by the second. Explain the steps.

5a3; a3
Answer:

The division is as follows:



Here when a3 is multiplied by 5 we get 5a3 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 5a3 becomes perfectly divisible by a3 and hence the quotient is 5 and remainder is 0.
Question 4.

Divide the first polynomial by the second. Explain the steps.

–30n2; 3n
Answer:

The division is as follows:



Here when 3n is multiplied by -10n we get -30n2 as the result.


Here the dividend is negative so the quotient has to be negative to get remainder zero.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So -30n2 becomes perfectly divisible by 3n and hence the quotient is -10n and remainder is 0.
Question 5.

Divide the first polynomial by the second. Explain the steps.

15m3; 5m
Answer:

The division is as follows:



Here when 5m is multiplied by 3m2 we get 15m3 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 15m3 becomes perfectly divisible by 5m and hence the quotient is 3m2 and remainder is 0.
Question 6.

Divide the first polynomial by the second. Explain the steps.

2b5; b2
Answer:

The division is as follows:



Here when b2 is multiplied by 2b3 we get 2b5 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 2b5 becomes perfectly divisible by b2 and hence the quotient is 2b3 and remainder is 0.
Question 7.

Divide the first polynomial by the second. Explain the steps.

78x4; –7x2
Answer:

The division is as follows:



Here when -7x2 is multiplied by -4x2 we get -28x4 as the result.


Here the divisor is negative so the quotient has to be negative to get remainder zero.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So -28x4 becomes perfectly divisible by -7x2 and hence the quotient is -4x2 and remainder is 0.
Question 8.

Divide the first polynomial by the second. Explain the steps.

–8y5; –y3
Answer:

The division is as follows:



Here when –y3 is multiplied by 8y2 we get -8y5 as the result.


Here the divisor is negative and the dividend is also negative so the quotient has to be positive to get remainder zero.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So -8y5 becomes perfectly divisible by –y3and hence the quotient is 8y2 and remainder is 0.
Question 9.

Divide.

4n4 ÷ n2
Answer:

The division is as follows:



Here when n2 is multiplied by 4n2 we get 4n4 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 4n4 becomes perfectly divisible by n2 and hence the quotient is 4n2 and remainder is 0.
Question 10.

Divide.

52m3 ÷ 13m
Answer:

The division is as follows:



Here when 13m is multiplied by 4m2 we get 52m3 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 52m3 becomes perfectly divisible by 13m and hence the quotient is 4m2 and remainder is 0.
Question 11.

Divide.

20c2 ÷ 15c
Answer:

The division is as follows:



Here when 15c is multiplied by 1c we get 15c2 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 20c2 when divided by 15c, the quotient is 1c and remainder is 5c2.
Question 12.

Divide.

36x5 ÷ 4x3
Answer:

The division is as follows:



Here when 4x3 is multiplied by 9x2 we get 36x5 as the result.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 36x5 becomes perfectly divisible by 4x3 and hence the quotient is 9x2 and remainder is 0.
Question 13.

Divide.

40a2 ÷ (–10a)
Answer:

The division is as follows:



Here when -10a is multiplied by -4a we get 40a2 as the result.


Here the divisor is negative so the quotient has to be negative to get remainder zero.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So 40a2 becomes perfectly divisible by -10a and hence the quotient is -4a and remainder is 0.
Question 14.

Divide.

(–33y4) ÷ (–11y)
Answer:

The division is as follows:



Here when –11y is multiplied by 3y3 we get -33y4 as the result.


Here the divisor is negative and the dividend is also negative so the quotient has to be positive to get remainder zero.


In division the sign of the product [divisor × quotient] is always inverted to get the final result.


So the sign of product should always be opposite to that of the dividend.


So -33y4 becomes perfectly divisible by -11y and hence the quotient is 3y3 and remainder is 0.
Question 15.

Write five different pairs of dividends and divisors such that the quotient in each case is 2x.
Answer:

To get quotient as 2x,


Let us use the dividend, divisor and quotient rule.


Dividend = Divisor × Quotient + Remainder


Here,


Remainder = 0


Quotient = 2x


1st Dividend, Divisor:


Let Divisor = 5x


∴ Dividend = 5x × 2x = 10x2


2nd Dividend, Divisor:


Let Divisor = 6x2


∴ Dividend = 6x2 × 2x = 12x3


3rd Dividend, Divisor:


Let Divisor = 7x3


∴ Dividend = 7x3 × 2x = 14x4


4th Dividend, Divisor:


Let Divisor = 8x


∴ Dividend = 8x × 2x = 16x2


5th Dividend, Divisor:


Let Divisor = 6x5


∴ Dividend = 6x5 × 2x = 12x6
Exercise 82
Question 1.

Fill in the blanks in the division shown below.





Quotient: …………….; Remainder: ………….
Answer:

(Law of indices, )




()




Quotient :


Remainder : 0


Here, the divisor is , dividend is


Dividend=(Divisor x Quotient)+(Remainder)


=



Question 2.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient: n


Remainder: 3
Question 3.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient: p


Remainder: -6
Question 4.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient: x-5


Remainder: 0
Question 5.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient: 2y+5


Remainder: 0
Question 6.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:



Quotient: 4a-6


Remainder: 0
Question 7.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient:


Remainder: 0
Question 8.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient:

Remainder: 0
Question 9.

Divide the binomial by the monomial. Write the quotient and the remainder.
Answer:


Quotient:

Remainder: 0
Question 10.

Divide.
Answer:


Quotient:


Remainder: 0
Question 11.

Divide.
Answer:


Quotient:


Remainder: 0
Question 12.

Divide.
Answer:


Quotient:


Remainder: 0
Question 13.

Divide.
Answer:


Quotient:


Remainder: 0
Question 14.

Divide.
Answer:


Quotient:


Remainder: 0
Question 15.

Divide.
Answer:


Quotient:


Remainder: 0
Question 16.

Divide.
Answer:


Quotient:


Remainder: 0
Question 17.

Divide.
Answer:


Quotient:

Remainder: 0
Question 18.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:

Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 19.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 20.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 21.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 22.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 23.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 24.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 25.

Divide. Write the answer in the form ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Exercise 83
Question 1.

Divide. Write the quotient and remainder.
Answer:


Quotient:


Remainder: 0
Question 2.

Divide. Write the quotient and remainder.
Answer:


Quotient:


Remainder: 8
Question 3.

Divide. Write the quotient and remainder.
Answer:


Quotient:


Remainder: 10
Question 4.

Divide. Write the quotient and remainder.
Answer:


Quotient:


Remainder: 13
Question 5.

Divide. Write the quotient and remainder.
Answer:



Quotient:


Remainder:



Question 6.

Divide. Write the quotient and remainder.
Answer:


Quotient:


Remainder: 6
Question 7.

Divide the trinomial by the monomial. Write the answer in the form, ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 8.

Divide the trinomial by the monomial. Write the answer in the form, ‘Dividend = divisor × quotient + remainder’.
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Exercise 84
Question 1.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 2.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:


Dividend:

Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 3.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:

The dividend can be rewritten as



Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 4.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:

The dividend can be rewritten as



Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 5.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 6.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:

The dividend can be rewritten as



Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 7.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:


Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 8.

Divide the polynomial by the binomial. Write your answer in the form: Divided = divisor × quotient + remainder
Answer:

The dividend can be rewritten as



Dividend:


Divisor:


Quotient:


Remainder:


‘Dividend = divisor × quotient + remainder’.
Question 9.

Divide the polynomial by each of the following binomials. Write the quotient and remainder.
Answer:


Quotient:
Remainder: 0
Question 10.

Divide the polynomial by each of the following binomials. Write the quotient and remainder.
Answer:


Quotient:
Remainder: 0
Question 11.

Divide the polynomial by each of the following binomials. Write the quotient and remainder.
Answer:


Quotient:Remainder: 0
Question 12.

Divide the polynomial by each of the following binomials. Write the quotient and remainder.
Answer:


Quotient:Remainder: 240
Question 13.

Divide the polynomial by each of the following binomials. Write the quotient and remainder.
Answer:


Quotient:Remainder: 0