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Division Of Algebraic Expressions

Class 8th Mathematics RD Sharma Solution
Exercise 8.1
  1. Write the degree of each of the following polynomials: (i) 2x^3 + 5x^2 - 7 (ii)…
  2. Which of the following expressions are not polynomiasl? (i) x^2 + 2x^-2 (ii)…
  3. Write each of the following polynomicals in the standard from. Also, write their…
Exercise 8.2
  1. 6x^3y^3z^2by3x^2yz Divide:
  2. 15m^3n^3by5m^2n^2 Divide:
  3. 24a^3b^3by-8ab Divide:
  4. -21abc^2by-7abc Divide:
  5. xyz^2by-9xz Divide:
  6. -72a^4 b^5 c^8 by -9a^2 b^2 c^3 Divide:
  7. 16m^3y^2/4m^2y Simplify:
  8. 32m^2n^2p^2/4mnp Simplify:
Exercise 8.3
  1. x+2x^2 + 3x^4 - x^5by2x Divide:
  2. y^4 - 3y^3 + 1/2 y^2by3y Divide:
  3. -4a^3 + 4a^2 + aby2a Divide:
  4. -x^6 + 2x^4 + 4x^3 + 2x^2by root 2x^2 Divide:
  5. 5z^3 - 6z^2 + 7zby2z Divide:
  6. root 3a^4 + 2 root 3a^3 + 3^2 - 6aby3a Divide:
Exercise 8.4
  1. 5x^3 - 15x^2 + 25xby5x Divide:
  2. 4z^3 + 6z^2 - zby - 1/2 z Divide:
  3. 9x^2y-6xy+12xy^2by - 3/2 xy Divide:
  4. 3x^3y^2 + 2x^2y+15xyby3xy Divide:
  5. x^3 + 7x+12byx+4 Divide:
  6. 4y^2 + 3y + 1/2 by2y+1 Divide:
  7. 3x^3 + 4x^2 + 5x+18byx+2 Divide:
  8. 14x^2 - 53x+45by7x-9 Divide:
  9. -21+71x-31x^2 - 24x^3by3-8x Divide:
  10. 3y^4 - 3y^3 - 4y^2 - 4ybyy^2 - 2y Divide:
  11. 2y^5 + 10y^4 + 6y^3 + y^2 + 5y+3by2y^3 + 1 Divide:
  12. x^4 - 2x^3 + 2x^2 + x+4byx^2 + x+1 Divide:
  13. m^3 - 14m^2 + 37m-26bym^2 - 12m+13 Divide:
  14. x^4 + x^2 + 1byx^2 + x+1 Divide:
  15. x^5 + x^4 + x^3 + x^2 + x+1byx^3 + 1 Divide:
  16. 14x^3 - 5x^2 + 9x-16y2x-1 Divide each of the following and find the quotient…
  17. 3x^3 - x^2 - 10x-3 by x-3 Divide each of the following and find the quotient…
  18. 6x^3 + 11x^2 - 39x-65by3x^2 + 13x+13 Divide each of the following and find the…
  19. 30x^4 + 11x^3 - 82x^2 - 12x+48by3x^2 + 2x-4 Divide each of the following and…
  20. 9x-4x^2 + 4by3x^2 - 4x+2 Divide each of the following and find the quotient and…
  21. Verify division algorithm i.e. Dividend=Divisor x Quotient + Remainder, in each…
  22. Divide 15y^4 + 16y^3 + 10/3 y-9y^2 - 6by3y-z Write down the coeficients of the…
  23. Using division of polynomials state whether (i) x+6 is a factor of x^2 - x-42 3…
  24. Find the value of a, if x+2 is a factor of 4x^4 + 2x^3 - 3x^2 + 8x+5a…
  25. What must be added to x^4 + 2x^3 - 2x^2 + x-1 so that the resulting polymonial…
Exercise 8.5
  1. Divide the first polynomial by the second polynomial in each of the following…
  2. Find Whether or not the first polynomial is a factor of the second: (i) x+1 ,…
Exercise 8.6
  1. x^2 - 5x+6byx-3 Divide:
  2. ax^2 - ay^2byax+ay Divide:
  3. x^4 - y^4byx^2 - y^2 Divide:
  4. acx^2 + (bc+ad) x+bdby (ax+b) Divide:
  5. (a^2 + 2ab+b^2) - (a^2 + 2ac+c^2) by2a+b+c Divide:
  6. 1/4 x^2 - 1/2 x-12 1/2 x-4 Divide:
Exercise 8.1
Question 1.

Write the degree of each of the following polynomials:

(i) 2x3 + 5x2 - 7

(ii)

(iii)

(iv)

(v)

(vi) 5

(vii)


Answer:

(i) 2x3 + 5x2 - 7


Degre is the highest power of the variable of a polynomial. In the given polynomial highest power is 3.


Therefore degree of the polynomial is 3.


(ii)


Degre is the highest power of the variable of a polynomial. In the given polynomial highest power is 2.


Therefore degree of the polynomial is 2.


(iii)


Degre is the highest power of the variable of a polynomial. In the given polynomial highest power is 2.


Therefore degree of the polynomial is 2.


(iv)


Degre is the highest power of the variable of a polynomial. In the given polynomial highest power is 7.


Therefore degree of the polynomial is 7.


(v)


Degre is the highest power of the variable of a polynomial. In the given polynomial highest power is 3.


Therefore degree of the polynomial is 3.


(vi) 5


Degre is the highest power of the variable of a polynomial. In the given polynomial there is no variable term.


Therefore degree of the polynomial is 0.


(vii)


Degre is the highest power of the variable of a polynomial. In the given polynomial highest power is 4.


Therefore degree of the polynomial is 4.
Question 2.

Which of the following expressions are not polynomiasl?

(i)

(ii)

(iii)

(iv)

(v)


Answer:

(i)


A polynomial never has negative or fractional power. In the given expression has negative power.


Therefore it is not a polynomial.


(ii)


A polynomial always has positive power.


Therefore the given expression is a polynomial.


(iii)


A polynomial always has positive power.


Therefore the given expression is a polynomial.


(iv)


A polynomial never has negative or fractional power. In the given expression has fractional power.


Therefore it is not a polynomial.


(v)


A polynomial never has negative or fractional power. In the given expression has negative power.


Therefore it is not a polynomial.
Question 3.

Write each of the following polynomicals in the standard from. Also, write their drgree:

(i)

(ii)

(iii)

(iv)

(v)

(vi)


Answer:

(i)


A polynomial in the standard form is written in the decreasing or increasing power of the variable.


Standard form of the polynomial: or


Degree is the highest power of the variable in the given expression.


Therefore degree of the polynomial is: 4


(ii)


A polynomial in the standard form is written in the decreasing or increasing power of the variable.


Standard form of the polynomial: or


Degree is the highest power of the variable in the given expression.


Therefore degree of the polynomial is: 6


(iii)


=


A polynomial in the standard form is written in the decreasing or increasing power of the variable.


Standard form of the polynomial: or


Degree is the highest power of the variable in the given expression.


Therefore degree of the polynomial is: 6


(iv)


=


A polynomial in the standard form is written in the decreasing or increasing power of the variable.


Standard form of the polynomial: or


Degree is the highest power of the variable in the given expression.


Therefore degree of the polynomial is: 6


(v)


=


A polynomial in the standard form is written in the decreasing or increasing power of the variable.


Standard form of the polynomial: or


Degree is the highest power of the variable in the given expression.


Therefore degree of the polynomial is: 6


(vi)


=


A polynomial in the standard form is written in the decreasing or increasing power of the variable.


Standard form of the polynomial: or


Degree is the highest power of the variable in the given expression.


Therefore degree of the polynomial is: 2
Exercise 8.2
Question 1.

Divide:
Answer:

= [Using an ÷ am = an-m]
Question 2.

Divide:
Answer:

= [Using an÷am = an-m]
Question 3.

Divide:
Answer:

= [Using an÷am = an-m]
Question 4.

Divide:
Answer:

= [Using an÷am = an-m] and [a° = 1]
Question 5.

Divide:
Answer:

= [Using an÷am = an-m] and [a° = 1]
Question 6.

Divide:

-72a4b5c8 by -9a2b2c3
Answer:


= 8a2b3c5
Question 7.

Simplify:
Answer:

= [Using an÷am = an-m]
Question 8.

Simplify:
Answer:

= [Using an÷am = an-m]
Exercise 8.3
Question 1.

Divide:
Answer:

= [Using an÷am = an-m]
Question 2.

Divide:
Answer:

= [Using an÷am = an-m]
Question 3.

Divide:
Answer:

= [Using an÷am = an-m]
Question 4.

Divide:
Answer:

= [Using an÷am = an-m]
Question 5.

Divide:
Answer:

= [Using an÷am = an-m]
Question 6.

Divide:
Answer:

[Using an÷am = an-m]
Exercise 8.4
Question 1.

Divide:
Answer:

= [Using an÷am = an-m]
Question 2.

Divide:
Answer:

= [Using an÷am = an-m]
Question 3.

Divide:
Answer:

= [Using an÷am = an-m]
Question 4.

Divide:
Answer:

= [Using an÷am = an-m]
Question 5.

Divide:
Answer:


Ans: x+3
Question 6.

Divide:
Answer:



Question 7.

Divide:
Answer:



Question 8.

Divide:
Answer:



Question 9.

Divide:
Answer:



Question 10.

Divide:
Answer:



Question 11.

Divide:
Answer:



Question 12.

Divide:
Answer:



Question 13.

Divide:
Answer:



Question 14.

Divide:
Answer:



Question 15.

Divide:
Answer:



Question 16.

Divide each of the following and find the quotient and remainder:
Answer:


Quotient:


Remainder: 4
Question 17.

Divide each of the following and find the quotient and remainder:

by


Answer:


Quotient:


Remainder: 39
Question 18.

Divide each of the following and find the quotient and remainder:
Answer:


Quotient:


Remainder: 0
Question 19.

Divide each of the following and find the quotient and remainder:
Answer:


Quotient:


Remainder: 0
Question 20.

Divide each of the following and find the quotient and remainder:
Answer:


Quotient:


Remainder: 0
Question 21.

Verify division algorithm i.e. Dividend=Divisor Quotient + Remainder, in each of the following. Also, write the quotient and remainder;
Answer:

(i)


Dividend = Divisor Quotient + Remainder


=


=


=


(ii)



Dividend = Divisor Quotient + Remainder


=


=


=


(iii)



Dividend = Divisor Quotient + Remainder


=


=


=


(iv)



Dividend = Divisor Quotient + Remainder


=


=


=


(v)



Dividend = Divisor Quotient + Remainder


=


=


=


(vi)



Dividend = Divisor Quotient + Remainder


=


=


=


(vii)



Dividend = Divisor Quotient + Remainder


=


=


=



Question 22.

Divide Write down the coeficients of the terms in the quotient.
Answer:


Quotient:


Coefficient of y3 = 5; Coefficient of y2 = Coefficient of y = Constant term = Coefficient of y2 =



Question 23.

Using division of polynomials state whether

(i) is a factor of 3

(ii) 4x-1 is a factor of

(iii) 2y-5 is a factor of

(iv)is a factor of

(v) is a factor of

(vi) is a factor of


Answer:

(i) is a factor of



Quotient:


Remainder: 0


Since remainder is 0 therefore is a factor of


(ii) 4x-1 is a factor of



Quotient:


Remainder: 15


Since remainder is 15 therefore is NOT a factor of


(iii) 2y-5 is a factor of



Quotient:


Remainder:


Since remainder is therefore is NOT a factor of


(iv)is a factor of



Quotient:


Remainder:


Since remainder is therefore is NOT a factor of


(v) is a factor of



Quotient:


Remainder:


Since remainder is therefore is a factor of


(vi) is a factor of



Quotient:


Remainder:


Since remainder is therefore is NOT a factor of



Question 24.

Find the value of a, if x+2 is a factor of


Answer:



Therefore substitute x = -2 in the given equation we get,
Question 25.

What must be added to so that the resulting polymonial is exactly divible by


Answer:


Quotient:


Remainder:


Therefore to be added.
Exercise 8.5
Question 1.

Divide the first polynomial by the second polynomial in each of the following Also write the quotient and remainder:

(i)

(ii)

(iii)

(iv)

(v)


Answer:

(i)



Quotient:


Remainder: 25


(ii)



Quotient:


Remainder:


(iii)



Quotient:


Remainder: 0


(iv)



Quotient:


Remainder: 5


(v)



Quotient:


Remainder: 2
Question 2.

Find Whether or not the first polynomial is a factor of the second:

(i)

(ii)

(iii)

(iv)

(v)

(vi)


Answer:

(i)



Quotient:


Remainder: 1


Since remainder is 1 therefore the first polynomial is NOT a factor of the second polynomial.


(ii)



Quotient:


Remainder: 56


Since remainder is 56 therefore the first polynomial is NOT a factor of the second polynomial.


(iii)



Quotient:


Remainder: 30


Since remainder is 30 therefore the first polynomial is NOT a factor of the second polynomial.


(iv)



Quotient:


Remainder: 0


Since remainder is 0 therefore the first polynomial is a factor of the second polynomial.


(v)



Quotient:


Remainder: 4


Since remainder is 4 therefore the first polynomial is NOT a factor of the second polynomial.


(vi)



Quotient:


Remainder: 2


Since remainder is 2 therefore the first polynomial is NOT a factor of the second polynomial.
Exercise 8.6
Question 1.

Divide:
Answer:


Quotient:


Remainder: 0
Question 2.

Divide:
Answer:


Quotient:


Remainder: 0
Question 3.

Divide:
Answer:


Quotient:


Remainder: 0
Question 4.

Divide:
Answer:


Quotient:


Remainder: 0
Question 5.

Divide:
Answer:


Quotient:


Remainder: 0
Question 6.

Divide:
Answer:


Quotient:


Remainder: 0