Class 8 RD Sharma Solutions Chapter 8 Division Of Algebraic Expressions  Exercise 8.6 (original) (raw)

Last Updated : 12 Sep, 2024

In this chapter, we explore the process of dividing algebraic expressions, a fundamental operation in algebra. We will examine techniques for simplifying expressions by dividing polynomials, factoring, and using long division methods, providing a solid foundation for more advanced algebra.

Understanding Algebraic Division

This section introduces the basic principles of dividing algebraic expressions, including key terminology and concepts. We will cover the foundational rules and properties that govern algebraic division, setting the stage for more complex operations.

Polynomial Long Division

Here, we delve into polynomial long division, a systematic method for dividing polynomials. Step-by-step examples will illustrate how to apply this technique to simplify expressions and solve algebraic problems.

Synthetic Division

Synthetic division offers a streamlined approach for dividing polynomials by linear factors. This section will explain the synthetic division process, highlighting its efficiency compared to traditional long division.

Factoring for Division

Factoring is a crucial skill for simplifying algebraic expressions before division. This section focuses on techniques for factoring polynomials, including methods such as factoring by grouping, special products, and the use of the greatest common factor (GCF).

**Question 1: Divide x 2 - 5x + 6 by (x - 3)

**Solution:

(x2 - 5x + 6)/(x-3)

Factorise the numerator and then divide it by (x-3):

x2 - 5x + 6

= x2 - 3x - 2x + 6

= (x2 - 3x - 2x + 6)/(x - 3)

= (x(x - 3) - 2(x - 3))/(x - 3)

= ((x - 3)(x - 2))/(x - 3)

= (x - 2)

Therefore, the answer is (x-2).

**Question 2: Divide ax 2 - ay 2 by (ax + ay)

**Solution:

(ax2 - ay2)/(ax + ay)

= a(x2 - y2)/(ax + ay)

= a(x - y)(x + y)/a(x + y)

= x - y

Therefore, the answer is (x - y).

**Question 3: Divide (x 4 - y 4) by (x 2 - y 2 )

**Solution:

(x4 - y4)/(x2 - y2)

= ((x2)2 - (y2)2)/(x2 - y2)

= ((x2 - y2) (x2 + y2)) / (x2 - y2)

= x2 + y2

Therefore, the answer is (x2 + y2).

**Question 4: Divide (acx 2 + (bc + ad)x + bd) by (ax + b)

**Solution:

(acx2 + (bc + ad)x + bd) / (ax + b)

= (acx2 + bcx + adx + bd) / (ax + b)

= (cx(ax + b) + d(ax + b)) / (ax + b)

= (ax + b)(cx + d) / (ax + b)

= cx + d

Therefore, the answer is (cx + d).

**Question 5: Divide (a 2 + 2ab + b 2 ) - (a 2 + 2ac + c 2 ) by (2a + b + c)

**Solution:

((a2 + 2ab + b2) - (a2 + 2ac + c2)) / (2a + b + c)

= ((a + b)2 - (a + c)2) / (2a + b + c)

= ((a + b + a + c)(a + b - a - c)) / (2a + b + c)

= (2a + b + c)(b - c) / (2a + b + c)

= b - c

Therefore, the answer is (b - c).

**Question 6: Divide ((1/4)x 2 **- (1/2)x - 12) by ((1/2)x - 4)

**Solution:

(1/4)x2 - (1/2)x - 12

= (1/4)(x2 - 2x - 48)

Now factorize it:

= (1/4)(x2 - 8x + 6x - 48)

= (1/4)(x(x - 8) + 6(x - 8))

= (1/4)(x - 8)(x + 6)

Now divide it by (1/2)x - 4:

= (1/4)(x - 8)(x + 6) / ((1/2)x - 4)

= (1/4)(x - 8)(x + 6) / (1/2)(x - 8)

= (1/4)(2/1)(x + 6)

= (1/2)x + 3

Therefore, the answer is (1/2)x + 3.

Summary

This exercise in RD Sharma's Class 8 textbook focuses on the division of algebraic expressions, particularly dealing with more complex polynomial divisions. It builds upon previous exercises, introducing students to advanced techniques for dividing polynomials by binomials and trinomials. The exercise emphasizes long division of polynomials, factoring methods, and the application of algebraic division in solving word problems and simplifying rational expressions.