Algebra Part 1 I Polynomials

 'Lesson 1'

Algebra Part I

[Polynomials, Remainder therom, Factor  therom]

Polynomials

Introduction of Polynomial Function

        A Polynomial is an algebra expression that represents the sum of a number of terms. In general the  nth degree polynomial of the x variable is denoted by                                                           

                                a₀ ^xn + a₁ ^xn-1 + a₂ ^xn-2 +.......+ a_n 

                                 [There n ϵ z+ and a₀ ≠ 0] 

  

"The maximum power of x, is called the polynomial's degree".       a0xn  is the leading term of this polynomial and, it's leading cofficient is a0

Example:-      2x5 + x4 + 3x3 +2x + 1  

  

this is a 5th digree polynomial. 

this leading term = 2x5

this leading cofficient = 2

     

Real Polynomials

              If real the all cofficients in a polynomial, (if a0, a1, a2, ....., an ϵ R) that polynomial's called real polynomial. 

According to the degree of a polynomial, we can divide the expressions into linear, quadratic, cubic. 

 

Polynomial functions 

If f(x) = a0 xn + a1 xn-1 + a2 xn-2 + .....+ an, then any value of x, f(x) have one value. there fore f(x) is a function of x. The polynomial function is defined for all real numbers. 

Therefore domain of   f = Df  = R

Polynomial Identities

If two polynomials are identical, the cofficients of the same number of terms must be the same.

Example:-      

             If, ax3 + ( b - 1 ) x2 + cx + d = 2x3 + 5x -1

                      a = 2  b = 1  c = 5  and  d = -1

1.1 Addition and Substraction of Polynomials

    Let's take f(x) and g(x) be the polynomial,  it's degrees 

respectively "m" and "n". The f(x) + g(x) defined as an addition of the same degree terms. Then it's results also a polynomial. If it's  m>n that polynomial's degree  is'm'.  If m=n "f(x) + g(x)" degree less than or equal to 'm'.

Let's look at few examples......

example 1:-  Add. 

                                    2x5 - 4x2 - 3 and 6x4 - 3x3 -2

                                    = 2x5 + 6x4 - 3x3 + (3 - 4) x (3 - 7)

                                    = 2x5+ 6x4- 3x3- x2 - 5

                                      =====================  

       

example 2:-  Substract. 

                           (4x3 - 4x2 - 2x + 3) - ( 2x3 - 3x + 7)

                                        = 4x3 - 4x2 - 2x + 3 - 2x3 - 3x + 7

                                        = (4 - 2) x3 - 4x2 + (3 - 2) x + (3 - 7)

                                        = 2x3 - 4x2 + x - 4

                         =================

 

example 3:-  Minimize.  

                      (4x6+ x -2 ) + (7x3 + 4x - 3) - (-2x6 + 5x - 2) 

                                = 4x6+ x -2  + 7x3 + 4x - 3 + 2x6 - 5x + 2

                                = (4 + 2) x6 + 7x3 + (1 + 4 - 5) x - (2 + 3 - 2) 

                                = 6x6 + 7x3 - 3

                    ===============