Algebraic Expressions

Notes of chapter: Algebraic Expressions are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 8.

 

Algebraic expression –

An algebraic expression is a mathematical expression that consists of terms [variables, constants and operations], exponentiation (raise to the power) and finding of roots in any combination.

E.g.- (i)3x + 5

It is an expression formed of

Variable x ,

Constant  5 and

Operation +

It is an expression formed of

Variable x ,

Constants 5 and

Operation +

Exponentiation x2

(1)Terms of an expression-

(a)It can be fixed value or a constant.

Eg- 5, 4, -10

Constant

(b)It can be a variable.

Eg:- x, y, z

Variable and coefficient

(c)It can be product of two or more variables.

Eg:- x2, xy, xyz

(d)It can be product of constant and variable.

Eg:– 5x, 6xy,2xy2,-3x

Hence, such parts of an expression which are formed separately are known as terms.

The expression which is formed with the use of operations (addition, subtraction, multiplication and division) on the variables and constants is called algebraic expression.

 

 Flow Chart of Algebraic Expression

(2)Types of terms

(i) Like terms-

When terms have same variable factors with same powers, they are called like terms.

Eg:– 2xy and 3xy are like terms.

(ii) Unlike terms-

When terms have different variable factors with different powers, they are called unlike terms.

Eg:- 2x and 2xy are unlike terms.

Factors of the terms

(3) Types of algebraic expressions-

(i)Monomial-

An expression with only one term is called a monomial.

Eg:- 2xy,-3, 2y2

(ii)Binomial-

An expression which contains two unlike term is called a binomial.

Eg:- 2xy – 3

(iii)Trinomial –

An expression which contains three unlike term is called a trinomial.

Eg:- 2xy – 3 + 2y2

(4)Addition and Subtraction of like terms-

The sum or difference of two or more like terms is a like term with a numerical coefficient equal to the sum or difference of the numerical coefficients of all the like terms.

Eg:-

(i)Add 5x and 2x

Ans-

5x + 2x

=(5+2)x[ distributive law]

=7x

(ii) Subtract 2x from5x.

Ans-

5x – 2x

= (5-2)x [distributive law]

=3x

(5)Addition and Subtraction of unlike terms-

Unlike terms can not be added or subtracted .So we write them as they are.

Eg:-

(i)Add 2x and 4y

Ans-

2x + 4y

(ii)Subtract 3x from 4y

Ans-

4y – 3x

(8) Multiplication of algebraic expressions-

(i) Steps for multiplication of monomial expressions-

Eg:- Multiply 2x and 3y

Step 1-

Multiply coefficients of monomial expressions.

Coefficients of product = Coefficient of first monomial × Coefficient of second monomial

2 × 3 = 6

Step 2-

Multiply algebraic factors of monomial expressions.

Algebraic factor of products

= algebraic factor of first monomial × algebraic factor of second monomial

Multiplication of x and y = xy

2x × 3y = 6xy

Some more examples-

(a) Multiplying two monomials-

(A) 3x × 4y

= 3 × 4xy

= 12xy

(B) x × 4y = 4xy

(C) 6x × (- 2y)

= 6 × (-2) × x × y

= -12xy

(D) 2x × 3x2

= 2 × 3 × x × x2

= 6 x3

(E)2x × (-2xyz)

= (2 × -2) × (x × xyz)

= -4x2yz

(b) Multiplying three or more monomials-

(A) 2x × 3y × 4z

= (2 × 3 × 4) × (x × y × z)

= 24xyz

(B)4xy × 2x2y2 × 4x3y3

= (4 × 2 × 4) × (x2 × x3) × (y2 × y3)

= 32x5y5

(ii)Multiplying a monomial by a polynomial-

(a) Multiplying a monomial by a binomial-

2x × (3y + 1)

= (2x × 3y) + (2x × 1) [Distributive Law]

= 6xy + 2x

(b) Multiplying a monomial by a trinomial-

2x × (x + 4xy – 2)

= (2x × x) + (2x × 4xy) – (2x × 2) [Distributive Law]

= 2x2 + 8x2y – 4x

(iii) Multiplying a polynomial by a polynomial-

(a) Multiplying a binomial by a binomial-

(x + y)(2x – 4y)

=x × (2x – 4y) + y × (2x – 4y)[Distributive law]

=(x × 2x) – (x × 4y) + (y × 2x) – (y × 4y)

= 2x2 – 4xy + 2xy – 4y2

= 2x2 – 2xy – 4y2

(b) Multiplying a binomial by a trinomial-

(2x + y)(x + y – 2)

=2x × (x + y – 2) + y (x + y – 2)[Distributive Law]

=(2x × x)+(2x × y) -(2x × 2) + (y × x) + ( y × y) –(y × 2)

=2x2+ 2xy -4x +xy + y2– 2y

=2x2+ 3xy -4x + y2– 2y

(9) Identity-

An identity is an equality which is true for every value of the variable in it.

Eg:-

(x + 1) (x + 2) = x2 + 3x + 2

is an identity because every value of variable(x) maintains equality.

If x = 2

Then,

(x + 1) (x + 2) = x2 + 3x + 2

(2 + 1)(2 + 2) = 22 + 3 × 2 + 2

3 × 4 = 4 + 6 +2

12 = 12

LHS = RHS

If x = 5

Then,

(x + 1) (x + 2) = x2 + 3x + 2

(5+ 1)(5 + 2) = 52 + 3 × 5 + 2

6 × 7 = 25 + 15 +2

42 = 42

LHS = RHS

Hence, equality is true for every value of variable.

(10) An equation is true for certain values of variable in it.

Eg:-

x2 + 3x + 2 = 12

If x = 2

LHS

=x2 + 3x + 2

= 22 + 3 × 2 + 2

= 4 + 6 +2

= 12

=RHS

If x = 5

LHS

=x2 + 3x + 2

= 52 + 3 × 5 + 2

= 25 + 15 +2

≠ 42

RHS

Hence, equation is true only for x = 2.

(11) Standard Identities-

(i) (a + b)2 = a2 + 2ab + b2

(ii) (a – b)2 = a2 – 2ab + b2

(iii)(a + b) (a – b) = a2 – b2

(iv) (x + a) (x + b) = x2 + (a + b)x + ab

(a) These four standard identities are useful in carrying out squares and products of algebraic expressions.

(b) These are also useful as alternative methods to calculate products of numbers and so on.

 

Helping Topics

NCERT Solutions Class 8

Worksheet Class 8

NCERT Solutions Class 7

Worksheet Class 7

Leave a comment