Page 161 - classs 6 a_Neat

Similarly. the perimeter p of a square of side s is given by the formula p = 4s.
5.POWERS OF A LITERAL Here 4 is a constant, whereas p ands are variables.
x × x is written as x , called x squared; REMARK In some situations literal numbers are also treated as constants. In such situations, it is pre
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x × x × x is written as x , called x cubed; sumed that the particular literal number will only take a fixed value.
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x × x × x × x is written as x , called x raised to the power 4; ALGEBRAIC EXPRESSION A combination of constants and variables connected by any one or
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x × x × x × x × x is written as x , called x raised to the power 5, and so on. more of the symbols +, -, x and + is called an algebraic expression.
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Instead of writing x , we write x only. The several parts of the expression, separated by the sign +or-, are called the
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In x , we call x the base and 5, the exponent or index. terms of the expression.
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Similarly, in y the base is y and the exponent is 7. Thus, (i) the expression 3x + 5y -2xyz has three terms, namely, 3x, 5y and -2xyz,
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EXAMPIE:1 Write the following using numbers, literals and signs of basic operations: (ii)the expression 5x -6x y + 8xy z -9 has four terms, namely, 5x , -6x y, 8xy z
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(i) 3 more than a number x and-9.
( ii) y less than 6
(iii) One-third of the sum of x and y Various types of algebraic expressions are as follows:
(iv) 5 less than the quotient of x by y (i) MONOMIALS: An expression which contains only one term is known as a monomial.
(v) The quotient of x by y added to the product of x and y Thus, 3x, 5xy , - 2abc, - 8, etc., are all monomials.
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(iv) 7 taken away from the sum of x and y (ii) BINOMIALS: An expression containing two terms is called a binomial.
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1 1 Thus, 6 - y, 2x + 3y, x -5xy z are all binomials.
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SOLUTION (i) x + 3 (ii) 6 - y (iii) ( + ) y (iv) 3 − 5 (iii) TRINOMIALS: An expression containing three terms is called a trinomial.
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x 3 Thus, 2 + x - y, a+ b + c, x - y + z , 6 + xyz+ x , etc., are all trinomials.
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(v) xy + (vi) x + y -7 (iv) QUADRINOMIALS: An expression containing four terms is called a quadrinomial.
y Thus, x + y + z - xyz, x + y + z + 3xyz, etc., are quadrinomials.
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EXAMPLE:2 Write the following statement using numbers, literals and signs of basic (v) POLYNOMIALS: An expression containing two or more terms is known as a polynomial.
operations: ‘7 times a number xis y less than a number z’. FACTORS When two or more numbers and literals are multiplied then each one of them is called a factor of
SOLUTION The given statement can be written as 7x = z - y. the product.
EXAMPLE:3 Write the following in the exponentialform: A constant factor is called a numerical factor, while a literal one is known as a literaljactor.
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(i)a × a × a × ... 12 times (ii) 5 × x ×x ×x × y × y (i) In 5x y, we have 5 as the numerical factor, whereas x and y are the literal factors.
SOLUTION We can write: (ii) In -xy, the numerical factor is -1 while x and y are literal factors.
(i)a × a × a× ... 12times=a . COEFFICIENTS In a product of numbers and literals, any of the factors ls called the coefficient
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(ii)5 × x × x × x × y× y = 5x y . of the product of other factors.
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EXAMPLE:4 Write down each of the following in the productform: EXAMPIE: 5 (i) In 5xy, the coefficient of y is 5x, and the coefficient of xis 5y.
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(i) a b (ii) 9b c (iii) 6a b c (ii)In -7 xy , the coefficient of x is -7y • and the coefficient of y is -7 x.
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SOLUTION We can write: (iii)In -x. the coefficient of x is -1.
(i) a b = a× a× b× b× b× b× b× b× b. CONSTANT TERM A term of the expression having no literal factor is called a constant term.
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(ii) 9b c = 9 × b× b× b× c. (i) In the expression 3x + 5, the constant term is 5. 4
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(iii) 6a b c = 6 × a× a× b× b× b× c × c× c × c. (ii) In the expression x + y - the constant term is - 5
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ALGEBRAIC EXPRESSIONS LIKE TERMS The terms having the same literaljactors are called like or similar terms.
UNLIKE TERMS The terms not having the same literal factors are called unlike or dissimilar
terms.
VARIABLES AND CONSTANTS In algebra, we come across two types of symbols, namely, constants
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and variables. EXAMPIE:6 (i) In the expression 6x 2y + 5xy -Bxy - 7yx we have 6x y and -7yx as like
terms, whereas 5xy and -8xy are unlike terms.
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A symbol having a fixed numerical value is called a constant. And, a symbol which takes on various (ii)3y , - 5y , 80y are like terms.
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numerical values is known as a variable. (iii)5xy and 5x y are unlike terms.
Consider the following examples:
FINDING THE VALUE OF AN ALGEBRAIC EXPRESSION if the values of all ltteral s by numbers appear-
The diameter d of a circle of radius r is given by the formula d = 2r. ing in a given expression are known, on replacing these literals by their numerical values we obtain an arithme-
Here, 2 is a fixed number and, therefore, a constant, whereas the literal numbers d and r de tic expression which can be evaluated.
pend upon the size of the circle and, therefore, they may take on various values. So, d and r Thus, for the gtven numerical values of the literals, we obtain the corresponding value of the
are variables. algebraic expression. The process o f replacing the literals by their numerical values is called substitution.