Class 8 RD Sharma Solutions - Chapter 8 Division Of Algebraic Expressions - Exercise 8.3

Exercise 8.3 in Chapter 8 of RD Sharma's Class 8 mathematics textbook focuses on the division of algebraic expressions. This exercise builds upon previous concepts of algebraic operations and introduces students to more complex division problems involving polynomials and algebraic fractions. Students will learn to simplify expressions, identify common factors, and apply the rules of exponents in division.

Question 1. Divide x + 2x² + 3x⁴ – x⁵ by 2x

Solution:

Here,

(x + 2x² + 3x⁴ – x⁵) / 2x

x/2x + 2x²/2x + 3x⁴/2x – x⁵/2x

By using this formula ⇒ aⁿ/a^m = a^n-m

1/2 x^1-1 + x^2-1 + 3/2 x^4-1 – 1/2 x^5-1

1/2 + x + 3/2 x³ – 1/2 x⁴

Question 2. Divide y⁴ – 3y³ + 1/2y² by 3y

Solution:

Here,

(y⁴ – 3y³ + 1/2y²)/ 3y

y⁴/3y – 3y³/3y + (½)y²/3y

By using this formula ⇒ aⁿ/a^m = a^n-m

1/3 y^4-1 – y^3-1 + 1/6 y^2-1

1/3y³ – y² + 1/6y

Question 3. Divide -4a³ + 4a² + a by 2a

Solution:

Here,

(-4a³ + 4a² + a) / 2a

-4a³/2a + 4a²/2a + a/2a

By using this formula ⇒ aⁿ/a^m = a^n-m

-2a^3-1 + 2a^2-1 + 1/2 a^1-1

-2a² + 2a + 1/2

Question 4. Divide –x⁶ + 2x⁴ + 4x³ + 2x² by √2x²

Solution:

Here,

(–x⁶ + 2x⁴ + 4x³ + 2x²) / √2x²

-x⁶/√2x² + 2x⁴/√2x² + 4x³/√2x² + 2x²/√2x²

By using this formula ⇒ aⁿ/a^m = a^n-m

-1/√2 x^6-2 + 2/√2 x^4-2 + 4/√2 x^3-2 + 2/√2 x^2-2

-1/√2 x⁴ + √2x² + 2√2x + √2

Question 5. Divide -4a³ + 4a² + a by 2a

Solution:

Here,

(-4a³ + 4a² + a) / 2a

-4a³/2a + 4a²/2a + a/2a

By using this formula ⇒ aⁿ/a^m = a^n-m

-2a^3-1 + 2a^2-1 + 1/2a^1-1

-2a² + 2a + 1/2

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

Solution:

Here,

(√3a⁴ + 2√3a³ + 3a² – 6a) / 3a

√3a⁴/3a + 2√3a³/3a + 3a²/3a – 6a/3a

By using this formula ⇒ aⁿ/a^m = a^n-m

√3/3 a^4-1 + 2√3/3 a^3-1 + a^2-1 – 2a^1-1

1/√3 a³ + 2/√3 a² + a – 2

Summary

Exercise 8.3 reinforces students' understanding of algebraic division, emphasizing the importance of identifying common factors, applying exponent rules, and simplifying expressions. By practicing these problems, students develop crucial skills in manipulating algebraic expressions, which forms a foundation for more advanced mathematical concepts in higher grades.