Expressions

In algebra, our primary tools are variables and constants, which we combine in various ways to create algebraic –to be filled by the user expressions. Variables, represented by letters like x, y, a, b, etc., are in nature, meaning their values can vary.

In contrast, constants are values, such as 5, -3, 100, and so forth.

To craft algebraic expressions, we use operations like addition, subtraction , multiplication , and division.

For instance, consider the expression 5x + 7.

Here, we multiply the variable x by the constant and then add the constant .

Another example is 3a - 4b.

In this case, we multiply variable a by and variable b by , and then subtract the latter from the former.

These expressions demonstrate how we can blend constants and variables to represent various mathematical relationships.

Moreover, algebraic expressions can also be formed by combining variables with themselves or with other variables.

For example: x² - y² is an expression where each variable is squared

("Squared" in mathematics refers to the operation of multiplying a number or a variable by itself. The term "squared" comes from the geometric concept where a square's area is calculated) and then subtracted from each other.

Another example is 2xy:

where we multiply two different variables, x and y, together. These combinations offer more complexity and versatility in algebra, allowing us to model a wide range of mathematical scenarios.

(i)x2

Instruction

The expression x2 is obtained by multiplying the variable x by itself.

x × x =

Just as 4 × 4 is written as , we write x × x = . It is commonly read as x squared.

In the same manner, we can write: x × x × x =

Commonly x3 is read as 'x cubed'. Later, you will realise that x3 may also be read {div.inline(slot="legend")} 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)2y2

The expression 2y2 is obtained from y

2y2 = × × (Enter integer value first)

Here by multiplying y with y we obtain y2 and then we multiply y2 by the constant .

(iii)3x2−5

In (3x2- 5) we first obtain x2, and multiply it by 3 to get 3x2.

From 3x2, we 5 to finally arrive at 3x2- 5.

Describe how the following expressions are obtained: 7xy + 5, x2y, 4x2 – 5x.

Instruction

(i)7xy + 5

7xy + 5, we first obtain .

it by 7 to get 7xy.

And 5 to 7xy to get the expression 7xy + 5.

(ii)x2y

x2y, we obtain x2

it by y to get x2y.

(iii)4x2 - 5x

4x2 - 5x, we first obtain x2

it by 4 to get 4x2.

And obtain second term by x by constant 5.

From 4x2, we 5x to finally arrive at 4x2 - 5x.

Terms of an 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 of 4 and x and then 5 to it.

Similarly consider the expression (3x2+7y)

Here we first formed 3x2 separately as a product of , x and x. We then formed 7y separately as a product of 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 , x and x, and the term −3xy is a product of , x and y.

Terms are added to form expressions: Just as the terms 4x and 5 are to form the expression (4x + 5), the terms (4x2) and -3xy are to give the expression (4x2-3xy)

This is because 4x2 + (–3xy) = 4x2 – 3xy.

Note: The minus sign (–) is included in the term. In the expression (4x2- 3xy), we took the term as (–3xy) and not as (3xy). That is why we do not need to say that terms are 'added or subtracted' to form an expression, just 'added' is enough. terms are 'added or subtracted' to form an expression, just 'added' is enough.

Types of algebraic expressions

An expression with only one term is called a

For example: 7xy, – 5m 3z2 , 4 , etc.

An expression which contains two unlike terms is called a .

For example: x + y, m – 5, mn + 4m, a2- b2 are binomials. The expression 10pq is not a binomial; it is a monomial. The expression (a + b + 5) is not a binomial as it contains terms.

An expression which contains three terms is called a

For example: The expressions x + y + 7, ab + a +b, 3x2– 5x + 2, m + n + 10 are trinomials.

The expression ab + a + b + 5 is, however not a trinomial; it contains terms and not three. The expression x + y + 5x a trinomial as the terms x and 5x are like terms.

In general, an expression with one or more terms is called a .

Thus a monomial, a binomial and a trinomial are all polynomials.

Classify the following expressions as a monomial, a binomial or a trinomial: a, a + b, ab + a + b, ab + a + b – 5, xy, xy +5,5x2 – x + 2, 4pq – 3q + 5p,7, 4m – 7n + 10, 4mn + 7.

Instruction

An algebraic expression with just one term is called a monomial.

Here, , , and are monomials.

An algebraic expression with two dissimilar terms is called a binomial.

Here, , , and are binomials.

An algebraic expression with three terms is called a trinomial.

Here, ,, , and are trinomials.

State with reasons, which of the following pairs of terms are of like terms and which are of unlike terms:

(i) 7x, 12y (ii) 15x, –21x (iii) – 4ab, 7ba (iv) 3xy, 3x (v) 6xy2 9x2y (vi) pq2– 4pq2 (vii) mn2 10mn.

Solution :

S.No	Pair	Factors	Remarks
(i)	7x, 12y	7, x 12,y	The variables in the terms are different.
(ii)	15x, -21x	15, x-21,x
(iii)	– 4ab, 7 ba	– 4, a, b 7,b,a	Remember ab=ba
(iv)	3xy , 3x	3, x, y 3,x	The variable y is only in one term.
(v)	6xy2 , 9x2y	6, x, y, y 9, x, x, y	The variables in the two terms match, but their powers do not match.
(vi)	pq2 , -4pq2	1, p, q, q – 4, p, q, q	Note, numerical factor 1 is not shown

Following simple steps will help you to decide whether the given terms are like or unlike terms:

(i) Ignore the numerical coefficients. Concentrate on the algebraic part of the terms.

(ii) Check the variables in the terms. They must be the same.

(iii) Next, check the powers of each variable in the terms. They must be the same. Note that in deciding like terms, two things do not matter (1) the numerical coefficients of the terms and (2) the order in which the variables are multiplied in the terms

Algebraic Expressions

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Expressions

Terms of an Expression

Types of algebraic expressions

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Algebraic Expressions

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Glossary

Expressions

Terms of an Expression

Types of algebraic expressions