Algebraic expressions are formed from variables and constants. The expression 3y + 5 is formed from variable ‘y’ and constants 3 and 5. The value of variable can vary. Therefore the value of expression can change with the change in the value of variable.
Algebraic expressions that contain exactly one term are called monomials. Expressions which contain 2 terms is called binomial, an expression containing three terms is a trinomial and so on.
Generally, an expression containing one or more terms with variables having no negative exponents is called polynomial. In a polynomial number of terms can be 1 or more than one.
When the variables and power of those variables are same then the terms formed are like terms.
Operations like addition, subtraction, multiplication can be done in algebraic expression. While adding or subtracting expressions, first look for like terms, then add the coefficient of like terms, and after it handle the unlike terms.
While multiplying a monomial with polynomial we have to multiply every term of polynomial with monomial. In case of binomial and trinomial term multiply each term of binomial, trinomial with every term of polynomial. In multiplication of polynomials with polynomials, first look for the like terms and combine them, then multiply it.
An equation is true for certain values of variable, it is not true for all values of variables but an algebraic identity is an equality which is true for all values of the variables. Some of the standard Algebraic Identities are as follows:-
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.
These all algebraic identities are useful in finding out squares and products of algebraic expressions. These identities are obtained by multiplying two binomial terms with each other.
For identity (i) take LHS
(a + b)2 = (a + b) (a + b),
= a (a + b) + b (a + b),
= a2 + ab + ba + b2,
= a2 + 2ab + b2,
Thus, LHS = RHS
Similarly, we can prove other identities also.
An algebraic identity gives a simple alternative method of solving problems on multiplication of binomial expressions.