ALGEBRAIC EXPRESSIONS, IDENTITIES AND  FACTORISATION

Algebraic Expression

• Terms are formed by the product of variables and constants, e.g.

–3xy, 2xyz, 5x², etc.

• Terms are added to form expressions, e.g. –2xy + 5x².
• Expressions that contain exactly one, two and three terms  are called monomials, binomials and trinomials, respectively.
• In general, any expression containing one or more terms with non- zero coefficients (and with variables having non-negative exponents) is called a polynomial.
• Like terms are formed from the same variables and the powers of these variables are also the same. But coefficients of like terms need not be the same.
• There are number of situations like finding the area of rectangle, triangle, etc. in which we need to multiply algebraic expressions.
• Multiplication of two algebraic expressions is again an algebraic expression.
• A monomial multiplied by a monomial always gives a monomial.
• While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the monomial using the distributive law a ( b + c) = ab + ac.
  • In 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) using the distributive property.
  • An identity is an equality, which is true for all values of its variables in the equality, i.e. an identity is a universal truth.  
  • An equation is true only for certain values of its variables.
  • Some standard identities:
    • (a + b)² = a² + 2ab + b²
    • (a – b)2 = a2 – 2ab + b2
    • (a + b) (a – b) = a2 – b2
    • (x + a) (x + b) = x2 + (a + b) x + ab

Factorisation

• Representation of an algebraic expression as the product of two or more expressions is called factorisation. Each such expression is called a factor of the given algebraic expression.
• When we factorise an expression, we write it as a product of its factors. These factors may be numbers, algebraic (or literal) variables or algebraic expressions.
• An irreducible factor is a factor which cannot be expressed further as a product of factors. Such a factorisation is called an irreducible factorisation or complete factorisation.
• A factor which occurs in each term is called the common factor.
• The factorisation done by using the distributive law (property) is called the common factor method of factorisation.
• Sometimes, many of the expressions to be factorised are of the form or can be put in the form: a² + 2ab + b², a² – 2ab + b², a² – b² or x² + (a + b) x + ab. These expressions can be easily factorised using identities:

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

x² + (a + b) x + ab = (x + a) (x + b)

• In the division of a polynomial by a monomial, we carry out the division by dividing each term of the polynomial by the monomial.
• In the division of a polynomial by a polynomial, we factorise both the polynomials and cancel their common factors.