ALGEBRAIC EXPRESSION


An algebraic expression is the expression which is made up of two components ‘variables and constants’ along with algebraic operations like addition, subtraction, multiplications, divisions etc.

These expressions are made up of terms.

                                                        Ex-  5x  +  2y  +  6

In the above given example
  •  ‘x’ and ‘y’ are variables which have no their own proper value 
  • ‘5’ and ‘2’ are coefficient that are respectively used with ‘x’ and ‘y’
  •  ‘6’ is constant that is well defined
  • This expression has three terms 5x, 2y and 6

 In algebraic expressions all the operations are done with variables, symbols and  letters which have no its own proper value. It must have one variable and one operation to be algebraic expression.

Algebraic Expression
Algebraic Expression


Types of algebraic expressions

There are mainly three types of algebraic expressions on the basis of terms.

              1. Mononomial Expression
              2. Binomial Expression
              3. Polynomial Expression

1. Mononomial Expression : The algebraic expression which has only one term is known as                    Mononomial Expression.
                                       Ex-    9y2, 3X4, 2y etc.

2. Binomial Expression : The algebraic expression which has only two terms is known as Binomial        Expression
                                      Ex -  4x2 + 5, 5xy + 3z etc.

3. Polynomial Expression : The algebraic expression which has more than one term and has also one non-zero terms with non-negative integral exponents of a variable is known as Polynomial                  Expression.
                                      Ex –  ab  +  bc  +  ca etc.


Algebraic Expression
Algebraic Expression





More types of algebraic expressions

Besides Mononomial, Binomial and Polynomial there are algebraic expression also classified into two other types
         
            1. Numeric  Expression
            2. Variable Expression

1. Numeric Expression : Numeric Expression consists only Numeric operations not any variable is        known as Numeric Expression

                                                Ex – 4x3+5, 6/4 etc.

2. Variable Expression : The expression which contains both numeric and variable operations is            known as Variable Expression.

                                               Ex - 3xy + 4, 7ab-3 etc.



How to is derived/create algebraic Expression

As we know that we have stated above that algebraic expression contains constant, variables and algebraic operations (add, Subtraction, division etc.). For the creation of algebraic expression we will have to create a situation first then we will create an algebraic expression.

The situation is that the age of  Mrs. Kiran is four times of her son and the sum of their ages is 40. 

 Now we will create an algebraic expression by this situation.

Let the age of son = x then the age of Mrs. Kiran would be 4x
The expression is
4x + x = 40; where x is the age of Mrs. Kiran’s son
--------------------------------------------------------------


Is algebraic expression Polynomial ?

All algebraic expressions are not  Polynomial but all polynomials are algebraic expression. Because, polynomial has only variables and coefficient with operation add, subtract, multiply, division etc. but algebraic expression contains irrational numbers.

In Mathematics there are several formulas which are used in various types and lesson. Several times we have to use formulas if we could not recall in any case then we have to face major problems while examination.

We always use various types of algebraic formulas as given below but most of the students don’t know how these formulas are derived from or proved.
So let’s prove it.

Formula 1 : (a+b)2   =  a2  +  2ab  +  b2

                     (a+b)2 = (a+b)(a+b)
                                 
                                ⇒  axa  +  axb  +  bxa  +  bxb

                               ⇒  a2  +  2ab  +  b2
                              -------------------------------------

Formula 2 : (a-b)2 =   a2 - 2ab + b2

                    (a-b)2 =  (a-b)(a-b)
                     
                              ⇒axa - axb - bxa + bxb
                      
                              ⇒a2 - 2ab + b2
                                   ---------------------------------------------

Formula 3 : (a+b+c)2 = a2 + b2 + c2 + 2(ab  + bc + ac)

                    (a+b+c)2 = (a+b+c)(a+b+c)

                                   ⇒ axa  +  axb  +  axc  +  bxa  + bxb  +  bxc  +  cxa  +  cxb  +  cxc

                                   ⇒a2  +  ab  +  ac  +  ab  +  b2  +  bc  +  ac  +bc  +  c2

                                   ⇒ a2  +  b2  +  c2  +  2ab  +  2bc  +  2ac

                                   ⇒a2  +  b2  +  c2  +  2(ab  +  bc  +  ac)
                                 -------------------------------------------------

Formula 4 : (a+b)3 =  a3 + 3a2b + 3ab2 + b3

                    (a+b)3 = (a + b)(a + b)(a + b)                      [ We multiply first two (a + b)(a + b) ]
                   
                               ⇒  (axa  +  axb  +  bxa  +  bxb) (a  +  b)
                     
                               ⇒  (a2  +  ab  +  ab  +  b2) (a  +  b)

                               ⇒  (a2  +  2ab  +  b2) (a  +  b)

                               ⇒  a2xa  +  2abxa  +  b2xa  +  a2xb  +  2abxb  +  b2xb
                   
                               ⇒  a3  +  2a2b  +  ab2  +  a2b  +  2ab2  +  b3

                               ⇒  a3  +  3a2b  +  3ab2  +  b3
                                              or
                               ⇒  a3  +  b3  +  3ab(a+b)
                          --------------------------------------

Formula 5 : (a-b)3 =  a3 - 3a2b + 3ab2 - b3

                    (a-b)3 = (a-b)(a-b)(a-b)                         [ We multiply first two (a + b)(a + b) ]
                  
                              ⇒  { (axa  +  ax(-b)  -  bxa  -  bx(-b) } (a - b)
                    
                              ⇒  (a2 - ab - ab + b2) (a - b)

                              ⇒  (a2 - 2ab + b2) (a - b)

                              ⇒  a2xa  -  2abxa  + b2xa  +  a2x(-b)  -  2abx(-b)  +  b2x(-b)
                   
                              ⇒  a3  -  2a2b  +  ab2  -  a2b  +  2ab2  -  b3

                              ⇒  a3  -  3a2b  +  3ab2  -  b3
                                               or
                              ⇒  a3  -  b3  -  3ab(a-b)
                            --------------------------------

Some more useful  formulas

⇨  a3  +  b3  +  c3  -  3abc  =  (a  +  b  +  c)(a2  +  b2  +  c2  -  ab  -  bc  -  ca)

⇨  If  a  +  b  +  c  =  0, then  a3  +  b3  +  c3  =  3abc

⇨  a2  -  b2  =  (a  +  b)(a  -  b)

⇨  (a  +  b)2 – (a  -  b)2  =  4ab

⇨  (a  +  b  +  c  +  d)2  =  a2  +  b2  +  c2  +  d2  +  2(ab  +  ac  +  ad  +  bc  +  bd  +  cd)

⇨  If  x + 1/x  =  a  then 
  • x2  +  1/x2  =  a2  -  2
  • x3  + 1/x3   =  a3  -  3a
  •  x4  + 1/x4   =  a4  -  4a2  +  2
  •  x5  + 1/x5   =  a5  -  5a3  +  5a

⇨  If  x - 1/x  =  a  then

  • x2  -  1/x2   =  a2  +  2
  • x3  - 1/x3   =  a3  +  3a
  • x4  - 1/x4   =  a4  +  4a2  +  2
  • x5  - 1/x5   =  a5  +  5a3  +  5a