RD Sharma Solutions for Class 8 Math Chapter 8 Division Of Algebraic Expressions are provided here with simple step-by-step explanations. These solutions for Division Of Algebraic Expressions are extremely popular among class 8 students for Math Division Of Algebraic Expressions Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma Book of class 8 Math Chapter 8 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma Solutions. All RD Sharma Solutions for class 8 Math are prepared by experts and are 100% accurate.

Page No 8.11:

Question 1:

Divide 5x3 − 15x2 + 25x by 5x.

Answer:

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Question 2:

Divide 4z3 + 6z2z by − 12z.

Answer:

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Question 3:

Divide 9x2y − 6xy + 12xy2 by −32xy.

Answer:

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Question 4:

Divide 3x3y2 + 2x2y + 15xy by 3xy.

Answer:

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Question 5:

Divide x2 + 7x + 12 by x + 4.

Answer:

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Question 6:

Divide 4y2 + 3y + 12 by 2y + 1.

Answer:

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Question 7:

Divide 3x3 + 4x2 + 5x + 18 by x + 2.

Answer:

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Question 8:

Divide 14x2 − 53x + 45 by 7x − 9.

Answer:

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Question 9:

Divide −21 + 71x − 31x2 − 24x3 by 3 − 8x.

Answer:

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Question 10:

Divide 3y4 − 3y3 − 4y2 − 4y by y2 − 2y.

Answer:

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Question 11:

Divide 2y5 + 10y4 + 6y3 + y2 + 5y + 3 by 2y3 + 1.

Answer:

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Question 12:

Divide x4 − 2x3 + 2x2 + x + 4 by x2 + x + 1.

Answer:

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Question 13:

Divide m3 − 14m2 + 37m − 26 by m2 − 12m +13.

Answer:

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Question 14:

Divide x4 + x2 + 1 by x2 + x + 1.

Answer:

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Question 15:

Divide x5 + x4 + x3 + x2 + x + 1 by x3 + 1.

Answer:

Page No 8.11:

Question 16:

Divide 14x3 − 5x2 + 9x − 1 by 2x − 1 and find the quotient and remainder

Answer:


Quotient = 7x2 + x + 5Remainder = 4

Page No 8.11:

Question 17:

Divide 6x3x2 − 10x − 3 by 2x − 3 and find the quotient and remainder.

Answer:


Quotient = 3x2+ 4x + 1 Remainder = 0

Page No 8.11:

Question 18:

Divide 6x3 + 11x2 − 39x − 65 by 3x2 + 13x + 13 and find the quotient and remainder.

Answer:


Quotient = 2x-5Remainder =0



Page No 8.12:

Question 19:

Divide 30x4 + 11x3 − 82x2 − 12x + 48 by 3x2 + 2x − 4 and find the quotient and remainder.

Answer:

Quotient =10x2-3x-12Remainder= 0

Page No 8.12:

Question 20:

Divide 9x4 − 4x2 + 4 by 3x2 − 4x + 2 and find the quotient and remainder.

Answer:


 Quotient = 3x2 4x 2 and remainder = 0.

Page No 8.12:

Question 21:

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

Dividend Divisor
(i) 14x2 + 13x − 15 7x − 4
(ii) 15z3 − 20z2 + 13z − 12 3z − 6
(iii) 6y5 − 28y3 + 3y2 + 30y − 9 2y2 − 6
(iv) 34x − 22x3 − 12x4 − 10x2 − 75 3x + 7
(v) 15y4 − 16y3 + 9y2 − 103y + 6 3y − 2
(vi) 4y3 + 8y + 8y2 + 7 2y2 − y + 1
(vii) 6y5 + 4y4 + 4y3 + 7y2 + 27y + 6 2y3 + 1

Answer:

(i)

Quotient = 2x + 3
Remainder = -3
Divisor = 7x - 4
Divisor × Quotient + Remainder = (7x - 4) (2x + 3) - ​3 
                                                = 14x+ 21- 8- 12 - ​3 
                                                = 14x2 + 13x - 15
                                                = Dividend
Thus,
Divisor × Quotient + Remainder = Dividend
Hence verified.

(ii)

Quotient = 5z2+103z+11Remainder = 54Divisor = 3z-6Divisor × Quotient +Remainder = (3z-6) 5z2+103z+11+54                                                           = 15z3+10z2+33z-30z2-20z-66+54                                                           = 15z3-20z2+13z-12                                                           = DividendThus,Divisor × Quotient + Remainder = Dividend                                                            
Hence verified.

(iii)


Quotient = 3y3-5y+32
Remainder = 0
Divisor = 2y2 - 6
Divisor × Quotient + Remainder =
(2y2-6) 3y3-5y+32+0=6y5-10y3+3y2-18y3+30y-9=6y5-28 y3+3y2+30y-9
= Dividend
 
Thus, Divisor × Quotient + Remainder = Dividend
Hence verified.

(iv)

Quotient  = - 4x3 + 2x2 - 8x + 30
Remainder  = - 285 
Divisor  = 3x + 7
Divisor × Quotient + Remainder =  (3x + 7) (- 4x3 + 2x2 - 8x + 30) - 285 
                                                 = - 12x4 + 6x3 - 24x2 + 90- 28x3 + 14x2 - 56x + 210 - ​285
                                                 = - 12x 4 - 22x3 - 10x2 + 34x - 75
                                                 =  Dividend
Thus,
Divisor × Quotient + Remainder = Dividend
Hence verified.

(v)


Quotient =  5y3-2y2+53y
Remainder =  6
Divisor = 3y - 2
Divisor × Quotient  + Remainder = (3y - 2) (5y3 - 2y2 53y) + 6
                                                = 15y4-6y3+5y2-10y3+4y2-103y+6
                                                = 15y4-16y3+9y2-103y+6
                                                =  Dividend
Thus,
Divisor × Quotient + Remainder = Dividend
Hence verified.

(vi)

Quotient =  2y + 5
Remainder =  11y + 2
Divisor =  2y2 - y + 1
Divisor × Quotient + Remainder =  (2y2 - y + 1) (2y + 5)11y + 2
                                                =  4y3 +10y2 - 2y2 - 5y + 2y + 5 + 11y + 2
                                                =  4y3 + 8y2 + 8y + 7
                                                =  Dividend
Thus,
Divisor × Quotient + Remainder  = Dividend
Hence verified.

(vii)




Quotient = 3y2 + 2y + 2
Remainder = 4y2 + 25y + 4
Divisor = 2y3 + 1
Divisor × Quotient + Remainder = (2y3 + 1) (3y2 2y + 2)4y225y + 4
                                                = 6y54y44y33y22y + 4y225y + 4
                                                6y54y44y37y227y + 6
                                                = Dividend
Thus,
Divisor × Quotient + Remainder = Dividend
Hence verified.

Page No 8.12:

Question 22:

Divide 15y4 + 16y3 + 103y − 9y2 − 6 by 3y − 2. Write down the coefficients of the terms in the quotient.

Answer:


 Quotient = 
5y3 + (26/3)y2 + (25/9)y + (80/27)
Remainder = (- 2/27)
Coefficient of y3 = 5
Coefficient
 of y2 = (26/3)
Coefficient of y = (25/9)
Constant = (80/27)

Page No 8.12:

Question 23:

Using division of polynomials, state whether
(i) x + 6 is a factor of  x2x − 42
(ii) 4x − 1 is a factor of 4x2 − 13x − 12
(iii) 2y − 5 is a factor of 4y4 − 10y3 − 10y2 + 30y − 15
(iv) 3y2 + 5 is a factor of 6y5 + 15y4 + 16y3 + 4y2 + 10y − 35
(v) z2 + 3 is a factor of z5 − 9z
(vi) 2x2x + 3 is a factor of 6x5x4 + 4x3 − 5x2x − 15

Answer:

(i)

Remainder is zero. Hence (x+6) is a factor of x2 -x-42
(ii)

As the remainder is non zero . Hence ( 4x-1) is not a factor of 4x2 -13x-12



(iii)




 The remainder is non zero,
 2y - 5 is not a factor of 4y4-10y3-10y2+30y-15.

(iv)

Remainder is zero.  Therefore, 3y2 + 5 is a factor of 6y5+15y4+16y3+4y2+10y-35.


(v)

Remainder is zero; therefore, z2 + 3 is a factor of z5 -9z.

(vi)



Remainder is zero ; therefore, 2x2-x+3 is a factor of 6x5-x4 +4x3-5x2-x-15.

Page No 8.12:

Question 24:

Find the value of a, if x + 2 is a factor of 4x4 + 2x3 − 3x2 + 8x + 5a.

Answer:

We have to find the value of a if (x+2) is a factor of (4x4+2x3-3x2+8x+5a).Substituting x=-2 in 4x4+2x3-3x2+8x+5a, we get:4(-2)4+2(-2)3-3(-2)2+8(-2)+5a=0or, 64-16-12-16+5a=0or, 5a=-20or, a=-4 If (x+2) is a factor of (4x4+2x3-3x2+8x+5a), a=-4.

Page No 8.12:

Question 25:

What must be added to x4 + 2x3 − 2x2 + x − 1 , so that the resulting polynomial is exactly divisible by x2 + 2x − 3?

Answer:


Thus, (- 2) should be added to (x4+2x3-2x2+x-1) to make the resulting polynomial exactly divisible by (x2+2x-3).



Page No 8.15:

Question 1:

Divide the first polynomial by the second in each of the following. Also, write the quotient and remainder:
(i) 3x2 + 4x + 5, x − 2
(ii) 10x2 − 7x + 8, 5x − 3
(iii) 5y3 − 6y2 + 6y − 1, 5y − 1
(iv) x4x3 + 5x, x − 1
(v) y4 + y2, y2 − 2

Answer:

(i) 3x2+4x+5x-2=3x(x-2)+10(x-2)+25(x-2)=(x-2)(3x+10)+25(x-2)=(3x+10)+25(x-2)Therefore, quotient=3x+10 and remainder=25.(ii) 10x2-7x+85x-3=2x(5x-3)-15(5x-3)+475(5x-3)=(5x-3)(2x-15)+475(5x-3)=(2x-15)+4755x-3Therefore, quotient=2x-15 and remainder=475.(iii) 5y3-6y2+6y-15y-1=y2(5y-1)-y(5y-1)+1(5y-1)(5y-1)=(5y-1)(y2-y+1)(5y-1)=(y2-y+1)Therefore, Quotient = y2-y+1 and remainder = 0

(iv) x4-x3+5xx-1=x3(x-1)+5(x-1)+5x-1=(x-1)(x3+5)+5x-1=(x3+5)+5x-1Therefore, quotient = x3+5 and remainder = 5.
(v) y4+y2y2-2=y2(y2-2)+3(y2-2)+6y2-2=(y2-2)(y2+3)+6y2-2=(y2+3)+6y2-2Therefore, quotient = y2+3 and remainder = 6.

Page No 8.15:

Question 2:

Find whether the first polynomial is a factor of the second.
(i) x + 1, 2x2 + 5x + 4
(ii) y − 2, 3y3 + 5y2 + 5y + 2
(iii) 4x2 − 5, 4x4 + 7x2 + 15
(iv) 4 − z, 3z2 − 13z + 4
(v) 2a − 3, 10a2 − 9a − 5
(vi) 4y + 1, 8y2 − 2y + 1

Answer:

(i) 2x2+5x+4x+1=2x(x+1)+3(x+1)+1x+1=(x+1)(2x+3)+1(x+1)=(2x+3)+1x+1 Remainder=1Therefore, (x+1) is not a  factor of 2x2+5x+4

(ii) 3y3+5y2+5y+2y-2=3y2(y-2)+11y(y-2)+27(y-2)+56y-2=(y-2)(3y2+11y+27)+56y-2=(3y2+11y+27)+56y-2 Remainder = 56 (y-2) is not a factor of 3y3+5y2+5y+2.


(iii)  4x4+2+154x2-5= x2(4x2-5)+3(4x2-5)+304x2-5= (4x2-5)(x2+3)+304x2-5=(x2+3)+304x2-5 Remainder = 30Therefore, (4x2-5) is not a factor of 4x4+7x2+15

(iv) 3z2-13z+44-z=3z2-12z-z+44-z=3z(z-4)-1(z-4)4-z=(z-4)(3z-1)4-z=(4-z)(1-3z)4-z=1-3z Remainder = 0 (4-z) is a factor of 3z2-13z+4.

(V) 10a2-9a-52a-3=5a(2a-3)+3(2a-3)+42a-3=(2a-3)(5a+3)+42a-3=(5a+3)+42a-3 Remainder = 4 ( 2a-3) is not a factor of 10a2-9a-5.

(vi) 8y2-2y+14y+1=2y(4y+1)-1(4y+1)+24y+1=(4y+1)(2y-1)+24y+1=(2y-1)+24y+1 Remainder = 2 (4y+1) is not a factor of 8y2-2y+1.



Page No 8.17:

Question 1:

Divide:
x2 − 5x + 6 by x − 3

Answer:

x2-5x+6x-3=x2-3x-2x+6x-3=x(x-3)-2(x-3)(x-3)=(x-3)(x-2)(x-3)= x-2

Page No 8.17:

Question 2:

Divide:
ax2ay2 by ax + ay

Answer:

ax2-ay2ax+ay=a(x2-y2)a(x+y)=a(x+y)(x-y)a(x+y)= x-y

Page No 8.17:

Question 3:

Divide:
x4y4 by x2y2

Answer:

 x4-y4x2-y2=(x2)2-(y2)2(x2-y2)=(x2+y2)(x2-y2)(x2-y2)= x2+y2

Page No 8.17:

Question 4:

Divide:
acx2 + (bc + ad)x + bd by (ax + b)

Answer:

acx2+(bc+ad)x+bd(ax+b)=acx2+bcx+adx+bd(ax+b)=cx(ax+b)+d(ax+b)(ax+b)=(ax+b)(cx+d)(ax+b)= cx+d

Page No 8.17:

Question 5:

Divide:
(a2 + 2ab + b2) − (a2 + 2ac + c2) by 2a + b + c

Answer:

(a2+2ab+b2)-(a2+2ac+c2)(2a+b+c)=(a+b)2-(a+c)2(2a+b+c)=(a+b+a+c)(a+b-a-c)(2a+b+c)=(2a+b+c)(b-c)(2a+b+c)=b-c

Page No 8.17:

Question 6:

Divide:
14x2-12x-12 by 12x-4

Answer:

14x2-12x-1212x-4=12x(12x-4)+3(12x-4)12x-4=(12x-4)(12x+3)(12x-4)=12x+3



Page No 8.2:

Question 1:

Write the degree of each of the following polynomials.
(i) 2x2 + 5x2 − 7
(ii) 5x2 − 3x + 2
(iii) 2x + x2 − 8
(iv) 12y7-12y6+48y5-10
(v) 3x3 + 1
(vi) 5
(vii) 20x3 + 12x2y2 − 10y2 + 20

Answer:

(i)  Correction  : It is 2x3+5x2-7  instead of 2x2+5x2-7.     The degree of the polymonial  2x3+5x2-7 is 3.(ii) The degree of the polymonial 5x2-35x+2 is 2.(iii) The degree of the polymonial 2x+x2-8 is 2.(iv) The degree of the polymonial 12y7-12y6+48y5-10 is 7.(v) The degree of the polymonial 3x3+1 is 3.(vi) 5 is a constant polynomial and its degree is 0.(vii) The degree of the polymonial 20x3+12x2y2-10y2+20 is 4.

Page No 8.2:

Question 2:

Which of the following expressions are not polynomials?
(i) x2 + 2x−2
(ii) ax+x2-x3
(iii) 3y35y + 9
(iv) ax1/2 + ax + 9x2 + 4
(v) 3x−2 + 2x−1 + 4x +5

Answer:

(i) x2+2x-2 is not a polynomial because -2 is the power of variable x is not a non negative integer.(ii) ax+x2-x3 is not a polynomial because 12 is the power of variable x is not a non negative integer.(iii) 3y3-5y+9 is a polynomial because the powers of variable y are non negative integers.(iv) ax12+ax+9x2+4 is not a polynomial because 12 is the power of variable x is not a non negative integer.(v) 3x-2+2x-1+4x+5 is not a polynomial because -2 and -1 are the powers of variable x are not non negative integers.

Page No 8.2:

Question 3:

Write each of the following polynomials in the standard form. Also, write their degree.
(i) x2 + 3 + 6x + 5x4
(ii) a2 + 4 + 5a6
(iii) (x3 − 1)(x3 − 4)
(iv) (y3 − 2)(y3 + 11)
(v) a3-38a3+1617
(vi) a+34a+43

Answer:

(i) Standard form of the given polynomial can be expressed as:(5x4+x2+6x+3) or (3+6x+x2+5x4) The degree of the polynomial is 4.(ii) Standard form of the given polynomial can be expressed as:(5a6+a2+4) or (4+a2+5a6) The degree of the polynomial is 6.(iii) (x3-1)(x3-4)=x6-5x3+4Standard form of the given polynomial can be expressed as:(x6-5x3+4) or (4-5x3+x6)The degree of the polynomial is 6.(iv) (y3-2)(y3+11)=y6+9y3-22Standard form of the given polynomial can be expressed as:(y6+9y3-22) or (-22+9y3+y6)The degree of the polynomial is 6.(v) (a3-38)(a3+1617)=a6+77136a3-617Standard form of the given polynomial can be expressed as:(a6+77136a3-617) or (-617+77136a3+a6)The degree of the polynomial is 6.(vi) (a+34)(a+43)=a2+2512a+1Standard form of the given polynomial can be expressed as:(a2+2512a+1) or (1+2512a+a2)The degree of the polynomial is 2.



Page No 8.4:

Question 1:

Divide 6x3y2z2 by 3x2yz.

Answer:

6x3y2z23x2yz=6×x×x×x×y×y×z×z3×x×x×y×z = 2x(3-2)y(2-1)z(2-1)=2xyz

Page No 8.4:

Question 2:

Divide 15m2n3 by 5m2n2.

Answer:

15m2n35m2n2=15×m×m×n×n×n5×m×m×n×n=3m(2-2)n(3-2)=3m0n1=3n

Page No 8.4:

Question 3:

Divide 24a3b3 by −8ab.

Answer:

24a3b3-8ab= 24×a×a×a×b×b×b-8×a×b=-3a(3-1)b(3-1)=-3a2b2

Page No 8.4:

Question 4:

Divide −21abc2 by 7abc.

Answer:

-21abc27abc= -21×a×b×c×c7×a×b×c=-3a(1-1)b(1-1)c(2-1)=-3c

Page No 8.4:

Question 5:

Divide 72xyz2 by −9xz.

Answer:

72xyz2-9xz=72×x×y×z×z-9×x×z=-8x(1-1)yz(2-1)=-8yz

Page No 8.4:

Question 6:

Divide −72a4b5c8 by −9a2b2c3.

Answer:

-72a4b5c8-9a2b2c3=-72×a×a×a×a×b×b×b×b×b×c×c×c×c×c×c×c×c-9×a×a×b×b×c×c×c=8a(4-2)b(5-2)c(8-3)=8a2b3c5

Page No 8.4:

Question 7:

Simplify:
16m3y24m2y

Answer:

16m3y24m2y=16×m×m×m×y×y4×m×m×y=4m(3-2)y(2-1)=4my

Page No 8.4:

Question 8:

Simplify:
32m2n3p24mnp

Answer:

32m2n3p24mnp=32×m×m×n×n×n×p×p4×m×n×p=8m(2-1)n(3-1)p(2-1)=8mn2p



Page No 8.6:

Question 1:

Divide x + 2x2 + 3x4x5 by 2x.

Answer:

x+2x2+3x4-x52x=x2x+2x22x+3x42x-x52x=12+x+32x3-12x4                                                                                   

Page No 8.6:

Question 2:

Divide y4-3y3+12y2 by 3y.

Answer:

y4-3y3+12y23y=y43y-3y33y+12y23y=13y(4-1)-y(3-1)+16y(2-1)=13y3-y2+16y

Page No 8.6:

Question 3:

Divide −4a3 + 4a2 + a by 2a.

Answer:

-4a3+4a2+a2a=-4a32a+4a22a+a2a=-2a(3-1)+2a(2-1)+12=-2a2+2a+12

                                                                       

Page No 8.6:

Question 4:

Divide -x6+2x4+4x3+2x2 by 2x2.

Answer:

-x6+2x4+4x3+2x22x2=-x62x2+2x42x2+4x32x2+2x22x2=-12x(6-2)+2x(4-2)+22x(3-2)+2x(2-2)=-12x4+2x2+22x+2                                            
                                            

Page No 8.6:

Question 5:

Divide 5z3 − 6z2 + 7z by 2z.

Answer:

5z3-6z2+7z2z=5z32z-6z22z+7z2z=52z(3-1)-3z(2-1)+72=52z2-3z+72
                                                                

Page No 8.6:

Question 6:

Divide 3 a4+23 a3+3a2-6a by 3a.

Answer:

3a4+23a3+3a2-6a3a=3a43a+23a33a+3a23a-6a3a=13a(4-1)+23a(3-1)+a(2-1)-2=13a3+23a2+a-2                                           
                                          



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