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
(b)It can be a variable.
Eg:- x, y, z
(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.
(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.
(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