We have already come across simple algebraic expressions like x + 3, y – 5, 4x + 5, 10y – 5 and so on. In Class VI, we have seen how these expressions are useful in formulating puzzles and problems. We have also seen examples of several expressions in the chapter on simple equations.
Expressions are a central concept in algebra. This Chapter is devoted to algebraic expressions. When you have studied this Chapter, you will know how algebraic expressions are formed, how they can be combined, how we can find their values and how they can be used
We now know very well what a variable is. We use letters x, y, l, m, ... etc. to denote variables. A variable can take various values. Its value is not fixed. On the other hand, a constant has a fixed value. Examples of constants are: 4, 100, –17, etc.
We combine variables and constants to make algebraic expressions. For this, we use the operations of addition, subtraction, multiplication and division. We have already come across expressions like 4x + 5, 10y – 20. The expression 4x + 5 is obtained from the variable x, first by multiplying x by the constant 4 and then adding the constant 5 to the product. Similarly, 10y – 20 is obtained by first multiplying y by 10 and then subtracting 20 from the product.
The above expressions were obtained by combining variables with constants. We can also obtain expressions by combining variables with themselves or with other variables. Look at how the following expressions are obtained:
x2, 2y2, 3x2 – 5, xy, 4xy + 7
(i) The expression x2 is obtained by multiplying the variable x by itself;
x × x = x2
Just as 4 × 4 is written as 42, we write x × x = x2. It is commonly read as x squared.
(Later, when you study the chapter ‘Exponents and Powers’ you will realise that x2
may also be read as x raised to the power 2).
In the same manner, we can write x × x × x = x3
Commonly, x3 is read as ‘x cubed’. Later, you will realise that x3 may also be read
as x raised to the power 3.
x, x2, x3, ... are all algebraic expressions obtained from x.
(ii) The expression 2y2 is obtained from y: 2y2 = 2 × y × y
Here by multiplying y with y we obtain y2 and then we multiply y2 by the constant 2.
(iii) In (3x2 – 5) we first obtain x2, and multiply it by 3 to get 3x2. From 3x2, we subtract 5 to finally arrive at 3x2 – 5.
(iv) In xy, we multiply the variable x with another variable y. Thus, x × y = xy.
(v) In 4xy + 7, we first obtain xy, multiply it by 4 to get 4xy and add 7 to 4xy to get the expression.
We shall now put in a systematic form what we have learnt above about how expressions are formed. For this purpose, we need to understand what terms of an expression and their factors are.
Consider the expression (4x + 5). In forming this expression, we first formed 4x separately as a product of 4 and x and then added 5 to it. Similarly consider the expression (3x2 + 7y). Here we first formed 3x2 separately as a product of 3, x and x. We then formed 7y separately as a product of 7 and y. Having formed 3x2 and 7y separately, we added them to get the expression.
You will find that the expressions we deal with can always be seen this way. They have parts which are formed separately and then added. Such parts of an expression which are formed separately first and then added are known as terms. Look at the expression (4x2 – 3xy). We say that it has two terms, 4x2 and –3xy. The term 4x2 is a product of 4, x and x, and the term (–3xy) is a product of (–3), x and y.
Terms are added to form expressions. Just as the terms 4x and 5 are added to form the expression (4x + 5), the terms 4x2 and (–3xy) are added to give the expression (4x2 – 3xy). This is because 4x2 + (–3xy) = 4x2 – 3xy.
We saw above that the expression (4x2 – 3xy) consists of two terms 4x2 and –3xy. The term 4x2 is a product of 4, x and x; we say that 4, x and x are the factors of the term 4x2. A term is a product of its factors. The term –3xy is a product of the factors –3, x and y.
We can represent the terms and factors of the terms of an expression conveniently and elegantly by a tree diagram. The tree for the expression (4x2 – 3xy) is as shown in the adjacent figure.
The factors are such that they cannot be further factorised. Thus we do not write 5xy as 5 × xy, because xy can be further factorised. Similarly, if x3 were a term, it would be written as x × x × x and not x2 × x. Also, remember that 1 is not taken as a separate factor
We have learnt how to write a term as a product of factors. One of these factors may be numerical and the others algebraic (i.e., they contain variables). The numerical factor is said to be the numerical coefficient or simply the coefficient of the term. It is also said to be the coefficient of the rest of the term (which is obviously the product of algebraic factors of the term). Thus in 5xy, 5 is the coefficient of the term. It is also the coefficient of xy. In the term 10xyz, 10 is the coefficient of xyz, in the term –7x2y2, –7 is the coefficient of x2y2.
When the coefficient of a term is +1, it is usually omitted. For example, 1x is written as x; 1 x2y2 is written as x2y2 and so on. Also, the coefficient (–1) is indicated only by the minus sign. Thus (–1) x is written as – x; (–1) x2 y 2 is written as – x2 y2 and so on.
Sometimes, the word ‘coefficient’ is used in a more general way. Thus we say that in the term 5xy, 5 is the coefficient of xy, x is the coefficient of 5y and y is the coefficient of 5x. In 10xy2, 10 is the coefficient of xy2, x is the coefficient of 10y2 and y2 is the coefficient of 10x. Thus, in this more general way, a coefficient may be either a numerical factor or an algebraic factor or a product of two or more factors. It is said to be the coefficient of the product of the remaining factors.
Identify, in the following expressions, terms which are not constants. Give their numerical coefficients:
xy + 4, 13 – y2, 13 – y + 5y2, 4p2q – 3pq2 + 5
|S. No.||Expression||Term (which is not a Constant)||Numerical Coefficient|
|1||xy + 4||xy||1|
|2||13 – y2||– y2||-1|
|3||13 – y + 5y2||
|4||4p2q – 3pq2 + 5||
When terms have the same algebraic factors, they are like terms. When terms have different algebraic factors, they are unlike terms. For example, in the expression 2xy – 3x + 5xy – 4, look at the terms 2xy and 5xy. The factors of 2xy are 2, x and y. The factors of 5xy are 5, x and y. Thus their algebraic (i.e., those which contain variables) factors are the same and hence they are like terms. On the other hand the terms 2xy and –3x, have different algebraic factors. They are unlike terms. Similarly, the terms, 2xy and 4, are unlike terms. Also, the terms –3x and 4 are unlike terms.
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