ALGEBRAIC EXPRESSIONS AND IDENTITIES- ESSENTIAL POINTS
Expressions are formed from variables and constants.
Constant: A symbol having a fixed numerical value. Example: 4, ½ , 4.5, etc.
Variable: A symbol which takes various numerical values. Example: x, y, z, etc.
Algebraic Expression: A combination of constants and variables connected by the sign +, -, × and ÷ is called algebraic expression.
Terms are added to form expressions. Terms themselves are formed as product of factors.
Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.
Like terms have same variables and same powers of these variables, while their coefficients may be different.
While adding (or subtracting) polynomials, first add or subtract like terms.
Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.
Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.
Trinomial: An expression containing three terms. Example: 2 2x + 3xy + 9 , 3x+2y+5z, etc.
Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents). A polynomial may contain any number of terms, one or more than one.
A monomial multiplied by a monomial always gives a monomial.
We multiply every term in the polynomial by the monomial while multiplying a polynomial by a monomial
In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).
An identity is an equality, which is true for all values of the variables in the equality. On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.
The following are the standard identities:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(a + b) (a – b) = a2 - b2
(y + a) (y + b) = y2 + (a + b)y + ab
Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.
Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.
Constant Term: A term of expression having no lateral factor.
Like term: The term having the same literal factors. Example 2xy and -4xy are like terms.
Unlike term: The terms having different literal factors. Example: 4x2 and 3xy are unlike terms.
Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. Numerical factor is the constant factor, while literal factor is a variable factor. The term 8x is the product of its factors 8 and x
Degree of polynomial is highest power of variables in the polynomial.
Degree of constant polynomial is always 0.
Product of positive and negative quantity is always negative.