Class-7 Algebraic Expressions

A collection of constants and variables connected by one or more mathematical expressions (addition, subtraction, multiplication, and division) is known as algebraic expression.

Examples: 2p², 5q³ − 2, p² + q² + 2pq,

Constant and Variables

In algebra we use two types of symbols, they are constants and variables. A Symbol which has a fixed value is known as Constant, and few examples are : 4, -3, 0, ⁴⁄₅, 2³⁄₅. A symbol which can have various numerical values is known as Variable(literal). We use the letters x, y, z, a, b, c, p, q, r etc. to denote a variable.

Properties of Algebraic Expressions

a + b = b + a        (Commutative law of addition)

(a + b) + c = a + (b + c)        (Associative law of addition)

a + 0 = 0 + a = a        (Additive identity)

a + (-a) = (-a) + a = 0        (Additive inverse)

ab = ba        (Commutative law of multiplication)

(ab) c = a (bc)        (Associative law of multiplication)

1 a = a 1 = a        (Multiplicative identity)

a(b + c) = ab + ca        (Distributive laws of multiplication)

Terms of an Algebraic Expressions

The various parts of an algebraic expression separated by +/- sign are known as terms of the algebraic expression.

Algebraic Expression	Number of Terms	Terms
2p	1	2p
3p + 5p²	2	3p, 5p²
5a − 7b	2	5a, -7b
3p² + 5p − 2	3	3p², 3, −2

Factors

Every constant or variable multiplied together to form a product is called a factor of the product. A constant factor is called a numerical factor and other factors are called variable factors.

Example:

a) In 5pq, the numerical factor is 5 and the variable factors are p and q.

b) In -3p²q, the numerical factor is -3 and the variable factors are p, p and q.

Coefficients

Any factor of variable term of an algebraic expression is called the coefficient of the remaining factor of the term. The constant part is called the numerical coefficient of the term and the remaining part is called the literal coefficient of the term. Let's see some examples.

Example 1. Find out all the coefficient of 5xy.

Solution. The numerical coefficient of 5xy = 5.

The literal coefficient of 5xy = xy.

The coefficient of x in 5xy = 5y.

The coefficient of y in 5xy = 5x.

Example 2. Find out all the coefficient of -7p²q.

Solution. The numerical coefficient of -7p²q = -7.

The literal coefficient of -7p²q = p²q. The coefficient of p = -7pq. The coefficient of q = -7p².

Like and Unlike Terms

The terms having same variable factors are known as like terms and the terms having different variable factors are known as unlike terms. Let's see some examples.

a) 2xy, 5xy, ⁴⁄₅xy, -9xy are like terms.

b) 2a²b, 7a²b, ³⁄₅a²b, -3a²b are like terms.

c) 3p, 5pq, 2p², 4p²q are unlike terms.

Type of Algebraic Expressions

Monomial expression
Binomial expression
Trinomial expression
Multinomial expression

Monomial Expression

An algebraic expression having only one term is known as monomial expression. Some examples are given below.

Example: 5p, 3a², ²⁄₅ab, 4a³b

Binomial Expression

An algebraic expression having two unlike terms is known as binomial expression. Some examples are given below.

Example: 2x + 3y, 2ab − 3a², 2p + 3pq

Trinomial Expression

An algebraic expression having three unlike terms is known as trinomial expression. Some examples are given below.

Example: 2x + 3y − 5, 2ab − 3a² + 2b², 2p + 3pq − p²

Multinomial Expression

An algebraic expression having two or more unlike terms is known as multinomial expression. Some examples are given below.

Example: 2ab − 3a², 2p + 3pq − p² + 5, 12xy² + 2x²y − 5xyz + 2z²

Polynomials in One Variable

An algebraic expression having only one variable is known as a polynomial in that variable. So, an algebraic expression of the form a + bx + cx² + dx³ + ..., where a, b, c, d, ... are constant and x is a variable, is known as a polynomial in the variable x. Let's see some examples.

Example 1. 2 + 3x is a polynomial in x of degree 1.

Example 2. 3a² + 5a − 3 is a polynomial in a of degree 2.

Example 3. 5p³ + 3p² + 10p is a polynomial in p of degree 3.

Polynomials in Two or More Variables

An algebraic expression having two or more variables is called a polynomial if the powers of every variable is a whole number. Let's see some examples.

Example 1. 3a² + 5b² − 3ab + 4 is a polynomial in two variables a and b. The degree of its terms are 2, 2, 1 + 1, 0. Here the degree of the polynomial is 2.

Example 2. 5p³ + 3q³ + 10p²q² − 6pq + 15 is a polynomial in two variables p and q. The degree of its terms are 3, 3, 2 + 2, 1 + 1, 0. Here the degree of the polynomial is 4.

Addition of Algebraic Expressions

To add algebraic expressions, we have to group like terms and then perform the addition on them. Let's see some examples.

Example 1. Add the below mentioned algebraic expressions.

        5p + 7q − 12r
        15r + 3q + 2p

Solution. (5p + 7q − 12r) + (15r + 3q + 2p)

        = 5p + 7q − 12r + 15r + 3q + 2p

        = 5p + 2p + 7q + 3q -12r + 15r

        = 7p + 10q + 3r

Example 2. Add the below mentioned algebraic expressions.

        5p³ + 3p² + 10p
        −2p³ − 2p² + 5p

Solution. (5p³ + 3p² + 10p) + (−2p³ − 2p² + 5p)

        = 5p³ + 3p² + 10p − 2p³ − 2p² + 5p

        = 5p³ − 2p³ + 3p² − 2p² + 10p + 5p

        = 3p³ + p² + 15p

Subtraction of Algebraic Expressions

To Subtract one expression from the other, we need to change the sign of each term of the expression to be subtracted. Then we have to group like terms and do the addition. Let's see some examples.

Example 1. Subtract 2p − 3q + 4r from 5p + 3q − 2r.

Solution. 5p + 3q − 2r − (2p − 3q + 4r)

        = 5p + 3q − 2r − 2p + 3q − 4r

        = 5p − 2p + 3q + 3q − 2r − 4r

        = 3p + 6q − 6r

Example 2. Subtract 2a² − 3a + 4 from 5a² + 6a − 2.

Solution. 5a² + 6a − 2 − (2a² − 3a + 4)

        = 5a² + 6a − 2 − 2a² + 3a − 4

        = 5a² − 2a² + 6a + 3a − 2 − 4

        = 3a² + 9a − 6