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Class 8 RD Sharma Solutions – Chapter 8 Division Of Algebraic Expressions – Exercise 8.1

  • Last Updated : 25 Jan, 2021

Question 1: Write the degree of each of the following polynomials:

(i) 2x3 + 5x2 – 7

(ii) 5x2 – 3x + 2

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(iii) 2x + x2 – 8

(iv) 1/2y7 – 12y6 + 48y5 – 10

(v) 3x3 + 1

(vi) 5

(vii) 20x3 + 12x2y2 – 10y2 + 20

Solution:

In a polynomial, degree is the highest power of the variable.

(i) 2x3 + 5x2 – 7



Given: 2x3 + 5x2 – 7

Therefore, the degree of the polynomial, 2x3 + 5x2 – 7 is 3.

(ii) 5x2 – 3x + 2

Given: 5x2 – 3x + 2

Therefore, the degree of the polynomial, 5x2 – 3x + 2 is 2.

(iii) 2x + x2 – 8

Given: 2x + x2 – 8

Therefore, the degree of the polynomial, 2x + x2 – 8 is 2.

(iv) 1/2y7 – 12y6 + 48y5 – 10

Given: 1/2y7 – 12y6 + 48y5 – 10

Therefore, the degree of the polynomial, 1/2y7 – 12y6 + 48y5 – 10 is 7.

(v) 3x3 + 1

Given: 3x3 + 1

Therefore, the degree of the polynomial, 3x3 + 1 is 3

(vi) 5

Given: 5

Therefore, the degree of the polynomial, 5 is 0 as 5 is a constant number.

(vii) 20x3 + 12x2y2 – 10y2 + 20

Given: 20x3 + 12x2y2 – 10y2 + 20

Therefore, the degree of the polynomial, 20x3 + 12x2y2 – 10y2 + 20 is 4.



Question 2: Which of the following expressions are not polynomials?

(i) x2 + 2x-2

(ii) √(ax) + x2 – x3

(iii) 3y3 – √5y + 9

(iv) ax1/2 + ax + 9x2 + 4

(v) 3x-2 + 2x-1 + 4x + 5

Solution:

(i) x2 + 2x-2

Given: x2 + 2x-2

Since variable x has a power of -2 which is negative and as a polynomial does not contain any negative powers or fractions.

Therefore, the given expression is not a polynomial.

(ii) √(ax) + x2 – x3

Given: √(ax) + x2 – x3

Since variable x has a power of 1/2 which is a fraction and as a polynomial does not contain any negative powers or fractions.

Therefore, the given expression is not a polynomial.

(iii) 3y3 – √5 y + 9

Given: 3y3 – √5 y + 9

Since the polynomial has positive powers i.e. non-negative integers.

Therefore, the given expression is a polynomial.

(iv) ax1/2 + ax + 9x2 + 4

Given: ax1/2 + ax + 9x2 + 4



Since variable x has a power of 1/2 which is a fraction and as a polynomial does not contain any negative powers or fractions.

Therefore, the given expression is not a polynomial.

(v) 3x-2 + 2x-1 + 4x + 5

Given: 3x-2 + 2x-1 + 4x + 5

Since variable x has a power of -2 and -1 which are negative and as a polynomial does not contain any negative powers or fractions.

The given expression is not a polynomial.

Question 3: Write each of the following polynomials in the standard from. 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 – 3/8) (a3 + 16/17)

(vi) (a + 3/4) (a + 4/3)

Solution:

(i) x2 + 3 + 6x + 5x4

Given: x2 + 3 + 6x + 5x4

Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(3 + 6x + x2 + 5x4) or (5x4 + x2 + 6x + 3)

The degree of the given polynomial is 4.

(ii) a2 + 4 + 5a6

Given: a2 + 4 + 5a6

Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(4 + a2 + 5a6) or (5a6 + a2 + 4)

The degree of the given polynomial is 6.

(iii) (x3 – 1) (x3 – 4)

Given: (x3 – 1) (x3 – 4)

x6 – 4x3 – x3 + 4

x6 – 5x3 + 4

Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.



Therefore, the expressions are:

(4 – 5x3 + x6) or (x6 – 5x3 + 4)

The degree of the given polynomial is 6.

(iv) (y3 – 2) (y3 + 11)

Given: (y3 – 2) (y3 + 11)

y6 + 11y3 – 2y3 – 22

y6 + 9y3 – 22

Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(-22 + 9y3 + y6) or (y6 + 9y3 – 22) 

The degree of the given polynomial is 6.

(v) (a3 – 3/8) (a3 + 16/17)

Given: (a3 – 3/8) (a3 + 16/17)

a6 + 16a3/17 – 3a3/8 – 6/17

a6 + (77/136)a3 – 48/136

Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(-48/136 + (77/136)a3 + a6) or (a6 + (77/136)a3 – 48/136)

The degree of the given polynomial is 6.

(vi) (a + 3/4) (a + 4/3)

Given: (a + 3/4) (a + 4/3)

a2 + 4a/3 + 3a/4 + 1

a2 + (25/12)a + 1

Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(1 + (25/12)a + a2) or (a2 + (25/12)a + 1)

The degree of the given polynomial is 2.




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