Polynomial – Addition and Multiplication

 

Motivation:

                      There are many applications for polynomial: the approximation to describe a graph, or the mathematical model for some daily problems. In data communications, some self-error correcting codes are based on some properties of special class of polynomials. It has been widely used in CD systems, cellular phones, and deep-space communication.

 

                    Usually, when we want to describe the expression associated with multiplication or addition on some variables, we use polynomials. A simple example is: if we know the length of  the side of a squared tile,  we would like to express the total area occupied by 1000 pieces of tiles. Furthermore, the work hours to put those tiles on the wall might depend on the area.  In this case, if we use x to stand for the length of the length of the tile, then the total area of 1000 pieces of tiles is

 

                                        1000x²

 

                    If the length of side changes, the total area also varies with this quantity. Furthermore, if the length of side is increased with 1 unit, what is the change associated with total area?  It can be formulated as

 

                                     1000(x+1)² – 1000x²

 

                  Thus, it could be easier to solve similar problems by developing some skills to tackle them.

 

 

Definition ( Polynomial with one variable)

             Let  c_0, c₁, c_2,  c_3, ..., c_n  be constants.   Then

                            c₀+  c₁x+c₂x² + c₃ x³ + … + c_nxⁿ  is a polynomial of  x.

 

             If  c_n is not 0, then we say that the order of this polynomial is n.

             Usually, we use  the notation

 

                               f(x) =  c₀ + c₁x+c₂x² + c₃ x³ + … + c_nxⁿ 

 

             and say: f(x) is a polynomial of x with order n. Or we say n is the degree of the polynomial .

            

             c_0, c₁, c_2,  c_3, ..., c_n are called the coefficients of the polynomial.

 

 

Example:  x² + 5x + 3 is a polynomial of x.

 

Example:  x³ + 2 is a polynomial of x .  And the order is 3.

 

Example: |x| is not a polynomial of x.

 

Example:     is not a polynomial of x.

 

Example:  3 is a polynomials of x.  Its order is 0.

 

Addition of Polynomials

            

                Addition of polynomials is the addition of the corresponding coefficients.

 

Ex1:  (x+2) + (5x+3)

         (x+2) + (5x+3 ) = (x+5x) + (2+3) = 6x + 5

 

        Sometimes, we use the following way :

 

                      x + 2

              +)   5x + 3

            ---------------------

                     6x + 5

 

 

                    

 

 

Multiplication of Polynomials

 

                Multiplication of  polynomials is governed by distributive law. 

 

               Ex1:    (x+2)x

                          (x+2)x = x² + 2x

 

               Ex2:   (x+2)(x+1)

                          (x+2)(x+1) = (x+2)x + (x+2)1

                                            = x² + 2x + x + 2

                                            = x² + 3x + 2

 

 

               We can use the following way to compute the result.

                         x  + 1

                  *)    x  + 2

             ---------------------

                         x² +  x                      (   (x+1)x  )

                                2x + 2                (  (x+1)2  )

            ------------------------

                         x² + 3x + 2

 

 

Division of Polynomials

 

                    For two polynomials  f(x) and g(x),  if we can find polynomials q(x) and r(x) such that

                                     f(x)=g(x)q(x) + r(x)

          and

                                      ( order of r(x) ) <  ( order of g(x) )

 

                then q(x) is the quotient polynomial of f(x) divided by g(x),

                         r(x)  is the remainder polynomial .

 

 

Example:   x² + 3x + 2 = (x+1)(x+2)

 

                  So,  when (x² + 3x + 2) is divided by (x+1),  the quotient polynomial is (x+2).

                   The remainder polynomial is 0.

 

 

                It can be done via the following procedure:

 

                                 x  + 2

                            -----------------------------------------

                    x+1  )   x²  + 3x + 2

                                x²  +   x

                           ----------------------------------

                                         2x + 2

                                         2x + 2

                                    ------------------

                                                  0

 

 

Example:    (x² + 3x + 5 ) divided by  (x+2)

 

                                   x   + 1

                              ----------------------------------------------

                      x+2  )   x²   +   3x  + 5

                                  x²    +  2x

                               --------------------------------

                                               x    + 5

                                               x    + 2

                                        -------------------

                                                         3

 

                         The quotient polynomial is  (x+1),  and the remainder polynomial is  3.