More  About  Polynomials  Syllabus  

1. Perform  division  of  polynomials  2. Understand  the  remainder  theorem    3. Understand  the  factor  theorem  4. Understand  the  concepts  of  the  greatest  common  divisor  and  the  least  common  multiple  of  polynomials  

5. Perform  addition,  subtraction,  multiplication  and  division  of  rational  function  

 

Polynomial  多項式  

Definition:  The sum or difference of terms which has variables raised to positive integer powers and which have coefficients that may be real or complex.

Standard Form: anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 • ak are the coefficients • akxk are called terms of the polynomial • a0 is the constant in the polynomial

Example: 5x3 – 2x2 + x – 13, x2y3 + xy, (1 + i)a2 +ib2 Polynomials with 1, 2 and 3 terms are called monomials, binomials and trinomials respectively.    

Which  of  the  followings  is/are  not  polynomial(s)?    

12   !!! + !"!!  

!!!! + !!! + !  

!   !!   log !    

Long  Division  of  polynomial  Long  division  of  polynomial  is  one  method  to  simplify  the  polynomial.      The  divider  is  usually  a  monomial  or  polynomial  whose  degree  is  less  then  that  of  the  dividend.    Long  division  of  polynomial  is  very  similar  to  long  numeric  division  we  study  in  primary  school.    Actually,  the  latter  can  be  regarded  as  one  specific  case  of  long  division  of  polynomial.    

Example  Divide  !! + 9! + 8  by  ! + 1   Divide  198  by  11  

                       Division  of  integer  by  integer  can  be  regarded  as  division  of  polynomial  with  the  variable  x  substituted  by  10!  

 

Exercise  1. Divide (x2 + 2x -15) by (x + 5) 2. Divide (2x5 - 5x4 + 7x3 +4x2 - 10x + 11) by (x3 + 2) 3. Divide the polynomial (3x3 - 11x2y + 11xy2 - 2y3)

by the binomial (x - 2y) 4. Divide (6x2 - 17x + 12) by (3x - 4) 5. Divide (10x5 + x3 + 5x2 - 2x - 2) by (5x2 - 2) 6. What is the remainder when (2y3 - y2 - 13y + 9) is divided by (y - 2)?

Remainder  Theorem  

Recall  your  memory  What  is  the  remainder  of  15÷ 7?  What  is  the  remainder  of  14÷ 7?  How  do  you  find  the  remainders  of  the  two  above  operations?  

Proof  Consider  f(x)  =  (x - r)q(x)  + R  Note  that  if  we  let  x = r,  the  expression  becomes  f(r)  =  (r - r)  q(r)  + R  Simplifying  gives:  f(r)  = R  This  leads  us  to  the  Remainder  Theorem,  which  states:  If  a  polynomial  f(x)  is  divided  by  (x  −  r)  and  a  remainder  R  is  obtained,  then  f(r)  = R.  

Example  ! ! = !! + 26! + 170   ! ! =  !! + 26! + 169  

 

Exercise  1. Find the remainder of f(x) = 3x2+ 5x − 8 divided by (x − 2) 2. Find the remainder of 3x3 − x2 − 20x + 5 divided by (x+4)  

Factor and Factor  Theorem  

Definition  of  factor  One  of  two  or  more  expressions  that  are  multiplied  together  to  get  a  product.  i.e. factors of 16 are 1, 2, 4, 8, 16 factors of ! + 3 ! + 4 are ! + 3  and  (! + 4)  

Proof  When f(x) is divided by g(x), by Division Algorithms,

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

If f(x) is divisible by g(x), we have !(!) ≡ 0, then f(x) = g(x) q(x) --- (*)

f(x) may have more than 1 factors, it equal to the difference between degree f(x) and degree g(x) +1,

i.e. deg(f(x))-deg(g(x))=1

Suppose that 1 factor of f(x) is (x – a), where a is constant, ! ! = ! − ! !!(!)

deg !! ! = deg ! ! − 1 Substitute x = a and q ! = ! − ! !! ! in (*)

! ! = ! − ! !! ! ! ! = 0   Hence, the Factor Theorem follows. (x – a) is a factor of polynomials f(x) whose degree ≥ 1, if and only if f(a) = 0

Example  ! ! = !! + 26! + 170   ! ! =  !! + 26! + 169  

 

Conclusion  If  f(r)  =  0,  then  (x-­‐r)  is  a  factor  of  the  polynomial  f(x).  If  f(r)  ≠  0,  then  f(r)  =  R  is  the  remainder  of  division  of  polynomial  f(x)  by  (x-­‐r).      

Exercise  1. Determine  whether  x-­‐2  is  a  factor  of  x2-­‐7x+10.    2. Determine  whether  x-­‐3  is  a  factor  of  x3-­‐3x2+4x-­‐12.  3. Show  that  x+1  is  a  factor  of  2x3+5x2-­‐9x-­‐12.  4. Find  a  so  that  x4+2x3-­‐ax2+x-­‐2  has  (x+2)  as  its  factor.  5. If  (x-­‐2)  and  (x-­‐3)  are  factors  of  x3+ax2+bx+12,  find  a  and  b.  6. Factorize  the  following  polynomials  using  factor  theorem.  

a) 2x2+x-­‐3 b) 5x2+6x-­‐8

Greatest  Common  divider  of  Polynomial  

Definition  [Numeric] The highest common factor of two positive integers is the largest integer, which is a factor of both. [Polynomial] The highest common factor of two polynomials p(x) and q(x) is that common divisor which has highest degree among all common divisors and in which the coefficient of highest degree term is positive.  PS.  Greatest  Common  Divider  and  Highest  Common  Factor  describe  the  same  thing!  

Recall  your  memory  What  is  the  HCF  of  63  and  14?  How  do  you  derive  it?  

Example  Find  the  HCF  of  36x2  -­‐  49  and  6x2  -­‐  25x  +  21  

If  (x  +  k)  is  the  HCF  of  (x2  +  ax  +  b)  and  (x2  +  cx  +  d),  then  find  the  value  of  k.  

Find  the  HCF  of  !! + 2! + 1  and  !! + 3!! + 3! + 1  Explain  why  !! + 2! + 1is  the  HCF  instead  of  monomial  (! + 1)  

Exercise  Find  the  HCF  of  the  following  polynomial  pairs.  

1. 2! − 3  !"#   ! − 2 ! 2! − 3 !  2.   !! − 4  !"#   ! + 2 !  3. ! + 1 !!"#   ! + 1 ! ! − 1  4. ! + 1 ! ! + 5 !!"#  !! + 10! + 25  

Least  Common  Multiple  of  Polynomial  

Definition  [Numeric] The least common multiple of two positive integers a and b is the smallest positive integer that is a multiple of both a and b. [Polynomial] The least common multiple of two or more polynomials is defined as the polynomial of the least degree, which is a multiple of all the given polynomials.

Recall  your  memory  What  is  the  LCM  of  14  and  42?  How  do  you  derive  it?  

Example  Find  the  LCM  of  the  pair  of  polynomials:  ! + 2 ! ! + 3  !"#   ! + 2 ! + 3 !  

Find  the  LCM  of  the  pair  of  polynomials:  ! 2! + 5  !"#   ! + 1 ! + 2 ! + 3  

 

Exercise  Find  the  LCM  of  the  following  pairs  of  polynomials  

1. ! + 2 !  !"#  !! + 8  2. ! + 1 !!"#   ! + 1 !(! − 1)  3. !! + 4! + 4  !"#  ! + 2  4. ! + 1 ! ! + 5 !!"#  !! + 10! + 25  

Relationship  of  HCF  and  LCM  What  is  the  H.C.F.  of  6  and  4?  What  is  their  L.C.M.?  You  know  very  well  that  the  H.C.F.of  6  and  4  is  2,  and  their  L.C.M.  is  12.  Now,  what  is  2  ×  12?    Is  it  not  the  product  6  ×  4?  Yes!  The  product  of  the  H.C.F.  and  the  L.C.M.  is  the  same  as  the  product  of  the  numbers.  The  same  result  is  true  even  for  polynomials.    

Product  of  two  given  polynomials  =  Product  of  their  H.C.F.  and  L.C.M.    

Proof  For  !! − 1  and  !! − 1  

Rational  Expression  

Definition  An  algebraic  expression,  which  is  the  quotient  of  two  polynomials,  is  called  a  rational  expression.  i.e.  !(!)

!(!)  where  ! ! and  ! !  are  polynomials.  

Examples:  !!!!!!

, !!!!!!!!!!!

, !!!!!!!!!!

, !!!!!

!!!!!    

 

Reduction  of  Rational  expressions    Reduce  the  rational  expression  into  the  lowest  form  

!! + 1!! + 3! + 2  

     

Reduce  the  rational  expression  into  the  lowest  form  

!! − 1(! − 1)(!! + 1)  

 

Exercise  

1. !!!!!!!!!!!!!!

 

2. (!!!!)(!!!)

(!!!!!!!)(!!!)

 

Operations  on  Rational  expressions  

Addition  and  Subtraction  Why  are  we  grouping  addition  and  subtraction  together?  It  is  because  subtraction  can  be  regarded  as  addition  of  negative  operands.  Add  !!!!

!!!  and   !!!

!!!   Subtract  !!!

!!!  from   !!!!

!!!!   Consider  addition  or  

subtraction  of  fractions:  27±

34  

 

Multiplication  and  Division  Why  are  we  grouping  multiplication  and  division  together?  It  is  because  division  can  be  regarded  as  multiplication  of  inversed  operands.  Multiply  !!!!

!!!!  !"#ℎ   !!!!

!!!   Multiply  !!!!

!!!  !"#ℎ   !!!

!!!   Consider  multiplication  

of  fractions:  314  ×  

521  

Divide  !!!!!!!

 !"   !!!!!!!

  Divide  !!!!!!!

 !"   !!!!!!

  Consider  division  of  fractions:  

109  ÷  

53  

Exercise