Have you remember Chapter 2 on Linear Equations in one variable? If you forgot, then it’s time to revise as we will build upon the Linear equations. Please go to this click to revise Linear Equations in One Variable”Link”. Let’s discuss what does it mean by the term expression. Expressions are formed from a combination of variables and constants. 3x+7, 2x^{2}+9, 8xy+x, 2x^{2} + 5y^{2} are few examples of expressions.
Let’s discuss the components of an expression by taking an example of expression 9x+10
Terms  In 9x+10 there are two terms namely 9x and 10. Hence terms are added to form expressions. Let’s take another example, In 6x  26 we have 2 terms namely 6x and 26.
Factors  Terms themselves can be formed as the product of factors, in expression 9x+10, the term 9x is the product of its two factors 9 and x. While term 5 has only one factor which is 5.
Coefficients  The numerical factor of a term is called coefficient OR in other words, we can say the number which is multiplied to variables is called coefficient. In the expression 9x+10, 9 is the coefficient of x.
What are Monomial, Binomial, Trinomial and Polynomial expressions?
Monomials  Expressions that have only one term. Example  4x, 5xy , 9z, 7xy^{2}, 54xyz etc.
Binomials  Expressions that have two terms. Example  x+y, 5xy+z, 9xy^{2} + xz, 61xy+xyz, etc
Trinomials  Expressions that have three terms. Example  x+y+z, 5xy+z+xy, 9xy^{2} + xz+99y, 61xy+xyz+x, etc
Polynomials  Expressions that have one or more terms with nonzero coefficients are polynomials. Example  4x, x+y+z, 5xy+z, 9xy^{2} + xz+99y+10x+7y^{2}+6yz, 61xy+xyz+73x+xz+53yz, etc
Let’s discuss another concept of Like and Unlike Terms.
Like Terms 
Unlike Terms 
Terms that have the same type of variables are known as Like Terms. Examples are given below. 
Terms which have different type of variables are Unlike Terms 



Operations on Algebraic Expressions
Addition > Only Like Terms are added when we add two expressions.
Example Add Expression 1 => 3x^{2} +7x  2 and Expression 2 => 4x  13
Let’s carry out addition of these two expressions.
You can see that the only like terms have been added
Subtraction > Just like addition, Only Like Terms are subtracted when we subtract one expression from another.
Example  Subtract 2x^2 + 7x from 3x^3 +9x^2 + 16x 22
Let’s carry out the subtraction operation.
As you can see only like terms have been subtracted
Multiplication > Multiplication operation is different from addition and subtraction and it can be applied between Like as well as, Unlike terms.
Type 1 Multiplication  Multiplying Monomials by Monomials
Example 1  Multiply 9a by 2a
=> 9ax2a
=> 9xax2xa (rewriting in product form)
=> 18xa^{2} (multiply number to number and variable to variable)
=> 18a^{2}
So product of 9a and 2a is 18a^{2}
Example 2  Multiply 4x^{2} , 16xy and 7yz
Solution: We need to simplify
Or, (rewriting in product form)
Or, (multiply number to number and variable to variable)
Or,
Or,
So the multiplication 4x^{2}, 16xy and 7z is
Type 2 Multiplication  Multiplying Monomials by Polynomials
Example 1  Multiply 3x by 7x+2y+z
=>
=> Use distributive law for solving this question.
=>
=>
Example 2  Multiply 2x+4y by 5x+3y+6z
=> Use distributive law for solving this question
=>
=>
=>
=>
=>
The product of 2x+4y and 5x+3y+6z is
Algebraic Identities
What is Identity?
An identity is an equality that holds true regardless of the values chosen for its variables. They are used in simplifying or rearranging algebra expressions. Or in simple words, an equality where LHS = RHS always is called an Identity.
Standard Identities  Identities which we use more often are called standard identities, They are mostly binomial or trinomial in nature. Let's discuss some standard identities.
Identity 1  (a+b)^{2} = a^{2} + b^{2} + 2ab
Let's derive this identity.
(a+b)^{2} = (a+b)x(a+b)
= a(a+b) + b(a+b)
= a^{2} +ab + ba +b^{2}
=> (a+b)^{2} = a^{2} + b^{2} + 2ab
Hence LHS = RHS for every value of a and b and this is an identity
Identity 2  (ab)^{2} = a^{2} + b^{2}  2ab
Let's derive this identity.
(ab)^{2} = (ab)x(ab)
= a(ab)  b(ab)
= a^{2} ab  ba +b^{2}
=> (a+b)^{2} = a^{2} + b^{2}  2ab
Hence LHS = RHS for every value of a and b and this is an identity
Identity 3  (ab)(a+b) = a^{2}  b^{2}
Let's derive this identity.
(ab)(a+b) = (ab)x(ab)
= a(a+b)  b(a+b)
= a^{2} +ab  ba b^{2}
=> (ab)(a+b) = a^{2}  b^{2}
Hence LHS = RHS for every value of a and b and this is an identity
Note  Identities 1,2 and 3 are known as standard identities
One more important identity which is very useful is (x+a)(x+b) = x^{2} + x(a+b) + ab
It's derivation is also similar. (x+a)(x+b) = x(x+b) + a(x+b)
= x2 + xb + ax + ab
= x2 + x(a+b) + ab
Hence LHS = RHS for every value of x, a and b and this is an identity
Let's learn to apply these identities
Example 1  Find (105)^{2 } ?
Solution : 105 can be written as (100+5)
=> (105)^{2} = (100+5)^{2}
Let's apply the identity 1, (a+b)^{2} = a^{2} + b^{2} + 2ab
Here a = 100 and b = 5
(100+5)^{2} = 100^{2} + 5^{2} + 2x100x5
= 10000 + 25 + 1000
= 11025
Example 2  Find (97)^{2 } ?
Solution : 97 can be written as (1003)
=> (97)^{2} = (1003)^{2}
Let's apply the identity 2, (ab)^{2} = a^{2} + b^{2}  2ab
Here a = 100 and b = 3
(1003)^{2} = 100^{2} + 3^{2}  2x100x3
= 10000 + 9  600
= 9409
Example 3  Find ( 3m + 2m)(3m  2m)
Let's apply the identity 3, (ab)(a+b) = a^{2}  b^{2}
( 3m + 2m)(3m  2m) = (3m)^{2}  (2m)^{2}
= 9m^{2}  4m^{2}