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

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

(i) 2x³ + 5x² – 7

(ii) 5x² – 3x + 2

(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) ax^1/2 + 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) ax^1/2 + ax + 9x² + 4

Given: ax^1/2 + 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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