ALGEBRAIC EXPRESSIONS AND IDENTITIES

INTRODUCTION
In the previous class, we have learnt about algebraic expressions and their addition and subtraction. Most of the expressions that we worked with had integer coefficients. In this chapter, we shall study multiplication of algebraic expressions in the form of monomials and binomials etc. Also, we shall learn to work with algebraic expressions that contain both integer and fractional coefficients. In other words, we shall work with algebraic expressions containing rational numbers as the coefficients of various terms. We shall also learn how to factorize algebraic expressions. But before all these things, we review here what we have learnt earlier.
6.2 REVIEW OF CONCEPTS AND DEFINITIONS
In algebra, we generally come across two types of symbols, namely constants and variables.
CONSTANT A symbol having a fixed numerical value is called a constant.
VARIABLE A symbol which takes various numerical values is called a variable.
ILLUSTRATION 1 We know that the perimeter P of a square of side s is given by P=4 x s. Here, 4 is a constant and P and s are

variables.

ILLUSTRATION 2

The perimeter P of a rectangle of sides 1 and b is given by P = 2(1 + b). Here, 2 is a constant and l and b are variables.

ALGEBRAIC EXPRESSIONS

A combination of constants and variables connected by the signs of fundamental operations of addition, subtraction, multiplication and division is called an algebraic expression.

TERMS

Various parts of an algebraic expression which are separated by the signs of + or — are called the ‘terms’ of the expression.
ILLUSTRATION 3 $2x^2$ — 3xy + $5y^2$ is an algebraic expression consisting of three terms, namely, $2x^2$ ,—3xy and $5y^2$ .
ILLUSTRATION 4 The expression $2x^3$ - $3x^2$ + 4x —7 is an algebraic expression consisting of four terms, namely, $2x^3$ ,- $3x^2$ ,4x and —7.
MONOMIAL An algebraic expression containing only one term is called a monomial.
ILLUSTRATION 5 —5, 3y, 7xy, x2yz, a2bc3 etc. are all monomials. Two monomials containing unlike terms when added give a binomial as defined below.
BINOMIAL An algebraic expression containing two terms is called a binomial.
ILLUSTRATION 6 The expressions 2x —3, 3x + 2y, xyz —5 etc. are all binomials.
Note that 3x + 7x is not a binomial, because 3x + 7x = lOx, which is a monomial.
TRINOMIAL An algebraic expression containing three terms is called a trinomial.
In other words, if three monomials are such that no two contain like terms, then their sum is a trinomial.
ILLUSTRATION 7 The expressions a—b+ $2x^2$ + $y^2$ —xy, $x^3$ — $2y^3$ - $2x^2 y^2z$ etc. are trinomials.
FACTORS Each term man algebraic expression is a product of one or more numbers (s) and / or literal(s). These number(s) and / or literal(s) are known as the factors of that term.
A constant factor is called a numerical factor, while a variable factor is known as a literal
factor.
COEFFICIENT In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.
ILLUSTRATION 8 In —5xy, the coefficient of x is —5y; the coefficient of y is —5x and the coefficient of xy is —5.
ILLUSTRATION 9 In —x , the coefficient of x is —1.
ILLUSTRATION 10 In $3a^2bc$ , the coefficient of $a^2$ is 3bc, the coefficient of b is $x3a^2c$ and the coefficient of c is $3a^2b$ .
CONSTANT TERM A term of the expression having no literal factor is called a constant term,
ILLUSTRATION 11 In the algebraic expression $x^2$ — xy + yz —4, the constant term is —4.
LIKE AND UNLIKE TERMS The terms having the same literal factors are called like or similar terms, otherwise they are called unlike terms.
ILLUSTRATION 12 In the algebraic expression $2a^2b$ + $3ab^2$ — 7ab — $4ba^2$ , we have 2 $2 a^2b$ and
— $4ba^2$ as like terms, whereas $3ab^2$ and —7ab are unlike terms.
EXERCISE 6.1
1. Identify the terms, their coefficients for each of the following expressions:
(i) $7x^2 yz$ —5xy (ii) $x^2$ +x+1 (iii) $3x^2 y^2$ — $5x^2 y^2 z^2$ + $z^2$
(iv) 9—ab+bc—ca (v) $\frac{a}{2}$ + $\frac{b}{2}$ – ab (vi) O.2x-0.3xy+0.5y
2. Classify the following polynomials as monomials, binomials, trinomials. Which
polynomials do not fit in any category?
(i) x + y (ii) 1000 (iii) x + $x^2$ + $x^3$ + $x^4$
(iv) 7 + a + 5b (v) 2b — $3b^2$ (vi) 2y — $3y^2$ + $4y^3$
(vii) 5x—4y+3x (viii) 4a- $15a^2$ (ix) xy+yz+zt+tx
(x) pqr (xi) $p^2q$ + $pq^2$ (xii) 2p+2q
ANSWERS
1. Terms Coefficients Terms Coefficients
(i) $7x^2yz$ 7 (ii) $x^2$ 1
-5xy -5 x 1
x 1
(iii) $3x^2 y^2$ 3 (iv) -ab -1
– $5x^2 y^2 z^2$ -5 bc 1
$z^2$ 1 9 9

(v) $\frac{a}{2}$ $\frac{1}{2}$
$\frac{b}{2}$ $\frac{1}{2}$
-ab -1
(vi) 0.2x 0.2
-0.3xy -0.3
0.5y 0.5
2. Monomial \ Binomial Trmomial None of these
(ii), (x) (i), (v), (viii) (xi), (xii) (iv), (vi), (vii) (iii), (ix)
6.2.1 ADDITiON OF ALGEBRAIC EXPRESSIONS
In adding algebraic expressions, we collect different groups of like terms and find the sum of like terms in each group. Note that the sum of several like terms is another like term. whose coefficient is the sum of the coefficients of those like terms.
Following examples will illustrate the same.
ILLUSTRATIVE EXAMPLES
Examplel Add:7×2 —4x+5,—3×2 +2x—land 5×2 —x+9.
Solution We have,
Required sum

ALGEBRAIC EXPRESSIONS AND IDENTITIES

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