Class 8 RD Sharma Solutions Chapter 8 Division Of Algebraic Expressions Exercise 8.1 (original) (raw)
Last Updated : 12 Sep, 2024
Exercise 8.1 in Chapter 8 of RD Sharma's Class 8 Mathematics textbook focuses on the fundamental concepts of dividing algebraic expressions. This exercise introduces students to the process of dividing monomials by monomials and polynomials by monomials. The problems are designed to help students understand the basic principles of algebraic division, including the laws of exponents and the distributive property. By working through these problems, students develop essential skills in simplifying and manipulating algebraic expressions, which form a crucial foundation for more advanced algebraic operations they will encounter in higher mathematics.
**Question 1: Write the degree of each of the following polynomials:
****(i) 2x** 3 + 5x 2 – 7
****(ii) 5x** 2 – 3x + 2
****(iii) 2x + x** 2 – 8
****(iv) 1/2y** 7 – 12y 6 + 48y 5 – 10
****(v) 3x** 3 + 1
****(vi) 5**
****(vii) 20x** 3 + 12x 2 y 2 – 10y 2 + 20
**Solution:
In a polynomial, degree is the highest power of the variable.
****(i)** **2x 3 + 5x 2 – 7
Given: 2x3 + 5x2 – 7
Therefore, the degree of the polynomial, 2x3 + 5x2 – 7 is 3.
****(ii)** **5x 2 – 3x + 2
Given: 5x2 – 3x + 2
Therefore, the degree of the polynomial, 5x2 – 3x + 2 is 2.
****(iii)** **2x + x 2 – 8
Given: 2x + x2 – 8
Therefore, the degree of the polynomial, 2x + x2 – 8 is 2.
****(iv)** **1/2y 7 – 12y 6 + 48y 5 – 10
Given: 1/2y7 – 12y6 + 48y5 – 10
Therefore, the degree of the polynomial, 1/2y7 – 12y6 + 48y5 – 10 is 7.
****(v)** **3x 3 + 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)** **20x 3 + 12x 2 y 2 – 10y 2 + 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) x** 2 + 2x -2
****(ii) √(ax) + x** 2 – x 3
****(iii) 3y** 3 – √5y + 9
****(iv) ax** 1/2 + ax + 9x 2 + 4
****(v) 3x** -2 + 2x -1 + 4x + 5
**Solution:
****(i)** **x 2 + 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) + x** 2 – x 3
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) 3y** 3 – √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 + 9x 2 + 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) x** 2 + 3 + 6x + 5x 4
****(ii) a** 2 + 4 + 5a 6
****(iii) (x** 3 – 1) (x 3 – 4)
****(iv) (y** 3 – 2) (y 3 + 11)
****(v) (a** 3 – 3/8) (a 3 + 16/17)
****(vi) (a + 3/4) (a + 4/3)**
**Solution:
****(i) x** 2 + 3 + 6x + 5x 4
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) a** 2 + 4 + 5a 6
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)** ****(x** 3 – 1) (x 3 – 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)** ****(y** 3 – 2) (y 3 + 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)** ****(a** 3 – 3/8) (a 3 + 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.
Summary
Exercise 8.1 covers a range of division problems involving algebraic expressions, primarily focusing on dividing monomials by monomials and polynomials by monomials. The questions progressively increase in complexity, starting with simple monomial divisions and advancing to more complex polynomial divisions. Students are expected to apply their knowledge of exponent rules, particularly when dividing terms with the same base. The exercise also reinforces the concept of the distributive property when dividing polynomials by monomials. Through these problems, students learn to simplify expressions, cancel common factors, and present their answers in the most reduced form. This exercise serves as a critical stepping stone in developing algebraic fluency and prepares students for more complex algebraic manipulations in future chapters.