Polynomial – HCF, LCM, and Polynomial Remainder Theorem

 

Motivation:

                     People are asking if polynomial has any application in “real” world.  Usually, the application of some knowledge you learn really depends on your creativity and imagination to apply the knowledge.  The more knowledge we accumulate, the more likely we can solve the problems in the future.

 

                     Polynomials have been widely used in data storage technology for the integrity of the data.  For example, in your CD, if it is slightly damaged, sometimes the data still can be recovered via some mathematical steps. That involves some special classes of polynomials.  Polynomials will be introduced here in more rigorous way than we have seen elsewhere earlier .

 

 

 

Definition ( Polynomial )

             Let  a₀ , a₁ , a₂, a₃, …, a_n be elements in a “field F”.  Then

       f(x) =   a₀ + a₁ x + a₂ x² + a₃ x³ + … + a_n xⁿ  is called a polynomial of x with coefficients defined in field F.   We just represent it as  f(x)  F[x] .

 

             Actually, the coefficients of polynomials are not necessarily defined in a “field”; it can be a “ring”.  But most of the time, people just put focus when those coefficients belong to a field.   It is not necessary to understand what is “ring” or “field” now.  We just bring in that concept to explain some other things.

 

             For the class of polynomials with coefficients of real numbers R, this class of polynomials is denoted as R[x] .  If we define the coefficients in integers Z, then we can denote that  f(x)  Z[x] .

             The operation for addition or multiplication depends on how you define “field”.

 

             The study of “field” or “ring” will be explored in advanced math class.  In high school, you just think the coefficients of a polynomial could be  Z(integer), Q(rational),  R(real), or C(complex).

 

Definition ( Factor )

                 f(x), q(x), r(x)    F[x] .  If  f(x) = q(x) r(x) ,  then r(x) and q(x) are called “factor” of f(x) .   We denote as r(x) | f(x),  and q(x) | f(x) . 

 

                 Please note that f(x), q(x), r(x) should belong to F[x],  in other words, the coefficients should be defined in the same “field” .    Let’s check the following example.

 

Example: f(x) = x² -2

                 If f(x) is defined in R[x], then (x-  )  is a factor of f(x).  But if you define f(x) in Q[x] , (x-  ) is not a factor of f(x).

 

                 Thus, if we speak of “polynomial factorization”,  it is necessary to specify the “field” for the coefficients.

 

 

 

 

Definition ( Irreducible Polynomial )

                If a polynomial f(x)  F[x] has no factor with lower degree other than constant,  then we say that this polynomial is irreducible in F.

 

                Please notice: when you say a polynomial is irreducible, you need to specify the “field”.   For example,  (x² -2 ) is irreducible in Z, but it is not irreducible in R.

 

 

 

 

Theorem ( Polynomial Remainder Theorem):

               f(x)  R[x] .   f(c) is the remainder polynomial  if you divide f(x) by (x-c) .

 

         Proof:

                        The degree of (x-c) is 1.  So, we know that the remainder polynomial is a constant.  Thus,  f(x) can be represented as

 

                                     f(x) = (x-c)q(x) + d

 

                          set x=c, we get

                                     f(c) = (c-c) q(c) + d = d

 

 

 

Example :  f(1)=1, f(2)=2. 

                 What is the remainder polynomial if  we divide f(x) by (x-1)(x-2) ?

 

        Sol:

                     Let  the remainder polynomial r(x)  be of the form

                                          ax+b   ( because (x-1)(x-2) is of order 2 )

                   

                     Thus,  we can set

                                 f(x) = (x-1)(x-2)q(x) + ax+b

 

    

 

                                   f(1) = a+b = 1

                                   f(2) = 2a + b = 2

 

                       So,   a = 1, b=0

                       The remainder polynomial is  x .

 

 

 

 

 

Definition ( Highest Common Factor , HCF )

                 f(x), g(x), r(x)  F[x] .   If   r(x)| f(x) and r(x)| g(x) , then r(x) is a common factor of f(x) and g(x) .   If  the degree of  r(x) is larger than or equal to any other common factors of f(x) and g(x),  then it is call Highest Common Factor ( HCF ). 

 

 

Example:      f(x)=2(x-1)³(x-2)(x-3)⁵

                     g(x)=6(x-1)²(x-3)⁶

 

                  Then,  HCF of f(x) and g(x)  is   (x-1)²(x-3)⁵  because there is no other common factor with higher order than this.   But consider the following:

 

                            2(x-1)²(x-3)⁵  is also a common factor of f(x) and g(x)

 

                 Usually, there exist multiple polynomials that are qualified as HCF. They only differ in a way by multiplying a constant .

 

 

 

Definition ( Least Common Multiplier, LCM )

              L(x), f(x), g(x)   F[x] .  If  f(x) | L(x) and g(x) | L(x),  then L(x) is called a common multiplier of f(x) and g(x).  If  the degree of L(x) is less than or equal to any other common multipliers of f(x) and g(x),  then L(x) is known as least common multiplier ( LCM ) of f(x) and g(x).

 

 

             Similar to HCF,   LCM is not unique.  There might exist many LCMs that only differ in a way by multiplying a constant.