RD SHARMA SOLUTION CHAPTER – 8 Division of Algebraic Expressions| CLASS 8TH MATHEMATICS-EDUGROWN

Exercise 8.1

Question 1.
Write the degree of each of the following polynomials :
(i) 2x³+5x²-7
(ii) 5x² – 3x +7 ’
(iii) 2x + x² –-8
(iv) 12 y⁷ -12y⁵ + 48y⁶ – 10
(v) 3x³ + 1
(vi) 5
(vii) 20x³ + 12x²y²– 10y² + 20
Solution:
(i) 2x³ + 5x²-7: The degree of this polynomial is 3.
(ii) 5x² – 3x + 2 : The degree of this polynomial is 2.
(iii) 2x + x² – 8 : The degree of this polynomial is 2.
(iv) 12 y⁷ – 12y⁶ + 48y⁵ – 10 : The degree of this polynomial is 7.
(v) 3x³ + 1 : The degree of this polynomial is 3.
(vi) 5 : The degree of this polynomial is 0 as it is only constant term
(vii) 20x³ + 12x²y² – 10y² + 20: The degree of this polynomial is 2 + 2 = 4.

Question 2.
Which of the following expressions are not polynomials :
(i) x² + 2x²                   
(ii) √a x + x²-x³
(iii) 3y³ – √5y + 9      
(iv) ax^1/2 + ax + 9x² + 4
(v) 3x² + 2x^-1 + 4x + 5
Solution:
(i) x² + 2x^-2 = x² + 2x 1×2 =x² + 1×2
: It is not xx polynomial as it has negative integral power.
(ii) √ax + x² – x³: It is polynomial.
(iii) 3y³  √5y + 9 : It is a polynomial.
(iv) ax^1/2+ ax + 9x² + 4: It is not a polynomial as  the degree of 1×2 is an integer.
(v) 3x² + 2x^-1 + 4x + 5 : It is not a polynomial as the degree of x^–2, x^-1 are negative.

Question 3.
Write each of the following polynomials in the standard form. Also write their degree.
(i) x² + 3 + 6x + 5x⁴
(ii) a¹ + 4 + 5a⁶
(iii) (x³ – 1) (x³ – 4)
(iv) (y³ – 2) (y³ + 11)
(v) (a3−38) (a3−1617)
(vi) (a+34) (a+34)
Solution:
Polynomial in standard form is the polynomial in ascending order or descending order.

Exercise 8.2

Question 1.
6x³y²z² by 3x²yz
Solution:

Question 2.
15m²n³ by 5m²n²
Solution:

Question 3.
24a³b³ by -8ab
Solution:

Question 4.
-21abc² by 7abc
Solution:

Question 5.
72xyz² by – 9xz
Solution:

Question 6.
-72a⁴b⁵c⁸ by – 9a²b²c³
Solution:

Simplify :

Question 7.

Solution:

Question 8.

Solution:

Exercise 8.3

Question 1.
x+2x²+3x⁴-x⁵ by 2x
Solution:

Question 2.
y⁴-3y³+ 12 y² by 3y
Solution:

Question 3.
-4a³ + 4a² + a by 2a
Solution:

Question 4.
-x⁶ + 2x⁴ + 4.x³ + 2x² by √2x²
Solution:

Question 5.
5z³ – 6z² + 7z by 2z
Solution:

Question 6.
√3 a⁴ + 2 √3 a³ + 3a² – 6a by 3a
Solution:

Exercise 8.4

Question 1.
5x³ – 15x² + 25x by 5x
Solution:

Question 2.
4z³ + 6z²-zby −12 z
Solution:

Question 3.
9x²y – 6xy + 12xy² by −32 xy
Solution:

Question 4.
3x²y² + 2x²y + 15xy by 3xy
Solution:

Question 5.
x² + 7x + 12 by x + 4
Solution:

Question 6.
4y² 4 + 3y + −12 by 2y + 1
Solution:

Question 7.
3x³ + 4x² + 5x + 18 by x + 2
Solution:

Question 8.
14x² – 53x + 45 by 7a – 9
Solution:

Question 9.
-21 + 71x – 31x² – 24ax³ by 3 – 8ax
Solution:

Question 10.
3y⁴ – 3y³ – 4y² – 4y by y² – 2y
Solution:

Question 11.
2y⁵ + 10y⁴ + 6y³ + y² + 5y + 3 by 2y³ + 1
Solution:

Question 12.
x⁴ – 2x³ + 2x² + x + 4 by x² + x + 1
Solution:

Question 13.
m³ – 14m² + 37m – 26 by m² – 12m + 13
Solution:

Question 14.
x⁴ + x² + 1 by x² + x + 1
Solution:

Question 15.
x⁵ + x⁴ + x³+x² + x+ 1 by x³ + 1
Solution:

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

Question 16.
14x³ – 5x² + 9x -1 by 2x – 1
Solution:

Question 17.
6x³ – x² – 10x – 3 by 2x – 3
Solution:

Question 18.
6x³+ 11x² – 39x – 65 by 3x² + 13x + 13
Solution:

Question 19.
30a⁴ + 11a³-82a²– 12a + 48 by 3a² + 2a- 4
Solution:

Question 20.
9x⁴ – 4x² + 4 by 3x² – 4x + 2
Solution:

Question 21.
Verify division algorithm i.e., Dividend = Divisor * Quotient + Remainder, in each of the following. Also, write the quotient and remainder :

Solution:

Question 22.
Divide 15y⁴ + 16y³ + 103 y – 9y² – 6 by 3y – 2
Write down the co-efficients of the terms in the quotient.
Solution:

Question 23.
Using division of polynomials state whether.
(i) x + 6 is a factor of x² – x – 42
(ii) 4x – 1 is a factor of 4x² – 13x – 12
(iii) 2y – 5 is a factor of 4y⁴ – 10y³ – 10y² + 30y -15
(iv) 3y + 5 is a factor of 6y⁵ + 15y + 16y + 4y+ 10y – 35
(v) z² + 3 is a factor of z⁵– 9z
(vi) 2x² – x + 3 is a factor of 60x⁵-x⁴ + 4x³ – 5x² -x- 15
Solution:

Question 24.
Find the value of ‘a’, if x + 2 is a factor of 4x⁴ + 2x³ – 3x² + 8x + 5a.
Solution:

Question 25.
What must be added to x⁴ + 2x³ — 2x² + x – 1 so that the resulting polynomial is exactly divisible by x² + 2x – 3.
Solution:

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) 3x² + 4x + 5, x – 2
(ii) 10x² – 7x + 8, 5x – 3
(iii) 5y³– 6y² + 6y-1,5y-1
(iv)x⁴-x³ + 5x,x-1
(v) y⁴ +y²,y²-2
Solution:
(i) 3x² + 4x + 5, x – 2
= 3x (x – 2) + 10x + 5
= 3x (x – 2) + 10 (x – 2) + 25
∴ Quotient = 3x + 10
Remainder = 25

(iii) 5y³ – 6y² + 6y – 1, 5y – 1
= y² (5y – 1) – 5y² + 6y- 1
= y² (5y – 1) -y (5y – 1) + 5y – 1
= y² (5y- 1) -y (5y- 1) + 1 (5y- 1)
∴ Quotient = y² – y + 1 and Remainder = 0
(iv) x⁴ – x³ + 5x, x – 1
= x³(x – 1) + 5x
= x³ (x – 1) + 5 (x – 1) + 5
∴ Quotient = x³ + 5, Remainder = 5
(v) y⁴+y²,y²– 2
= y²(y² – 2) + 3y²
= y² (y² – 2) + 3 (y² – 2) + 6
∴ Quotient =y² + 3 and Remainder = 6

Question 2.
Find, whether or not 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, 4.x⁴ + 7x² + 15
(iv) 4-z, 3z² – 13z + 4
(v) 2a-3,10a² – 9a – 5
(vi) 4y+1 ,8y²-2y + 1
Solution:
(i) x + 1, 2x² + 5x + 4
2x² + 5x + 4 = 2x (x + 1) + 3x + 4
= 2x (x + 1) + 3 (x + 1) + 1
∵ Remainder = 1
∴ x + 1 is not a factor of 2x² + 5x + 4
(ii) y – 2, 3y³ + 5y² + 5y + 2
3y³ + 5y² + 5y + 2 = 3y²(y – 2)+11y² + 5y + 2
= 3y²(y – 2)+11y (y – 2) + 27y + 2
= 3y² (y – 2) + 11y (y – 2) + 27 (y – 2) + 56
∵ Remainder = 56
∴ y – 2 is not a factor of 3y³ + 5y² + 5y + 2
(iii) 4x² – 5, 4x⁴ + 7x² + 15
4x⁴ + 7x² + 15 = x² (4x² – 5) + 12x² + 15
= x² (4x² – 5) + 3 (4x² – 5) + 30
∵ Remainder = 30
∴ 4x² – 5 is not a factor of 4x⁴ + 7x² + 15
(iv) 4 – z, 3z² – 13z + 4
3z² – 13z + 4 = -3z (-z + 4) – z + 4
= -3z (-z + 4) + 1 (-z + 4)
∵ Remainder = 0
∴ 4 – z or – z + 4 is a factor of 3z² – 13z + 4
(v) 2a – 3, 10a² – 9a – 5
10a² – 9a – 5 = 5a (2a – 3) + 6a – 5
= 5a (2a – 3) + 3 (2a – 3) + 4
∵ Remainder = 4
∴ 2a – 3 is not a factor of 10a² – 9a – 5
(vi) 4y + 1, 8y² – 2y + 1
8y² – 2y + 1 = 2y (4y + 1) – 4y + 1
= 2y (4y + 1) – 1 (4y + 1) + 2
∵ Remainder = 2
∴ 4y + 1 is not a factor of 8y² – 2y + 1

Exercise 8.6

Divide :

Question 1.
x² – 5x + 6 by x – 3
Solution:

Question 2.
ax² – ay² by ax + ay
Solution:

Question 3.
x⁴ – y⁴ by x² – y²
Solution:

Question 4.
acx² + (bc + ad)x + bd by (ax + b)
Solution:

Question 5.
(a² + 2ab + b²)- (a² + 2ac + c²) by 2a + b + c
Solution:

Question 6.
14 x² – 12 x- 12 by 12 x – 4
Solution: