How to find reminder and quotient of equation

Reminder Theorem

Statement: if f(x) is a polynomial of degree m in x, then the reminder in the division of f(x) by x-α is equal to f(x).

Proof: By Euclid’s or division algorithm we can find, the polynomial q(x) and r(x) such that f(x)=(x-α)q+r(x). Where either r(x)=0 or the degree of r(x) is less than g(x)(x-α).

Therefore r(x)=0 or the degree of r(x) is less than “1” or will be equal to “0” . In the second case r(x)=k(constant)

i) f(x)=(x-α)q(x)+k

  put x=α 

 (x)=(α-α)q(x)+k

 f(α)=k ;

Factor theorem

If “α”  is   the root f(x)=0 then x-α is  factor of f(x)

Proof:  f(x)=(x-α)q(x)+k --> (1)

Put x=α  , f(α)=k

Put f(α)=0 ,   “α” is a root of the equation f(x)=0 => k=0 ;

Substituting => f(x)=(x-α)q(x)

v  x-α is a factor of f(x) ;

Fundamental theorem of Algebra

Every polynomial equation of degree n≥ 1 has at least one or real or imaginary.

Synthetic  division for finding quotient and remainder for dividing a polynomial .

f(x)=a_0Xⁿ+a₂x^n-1   by x-h we apply a method called synthetic division.

Problems: 

1)  Find the quotient and reminder x⁴-5x³+6x²+7x-9   by x-1 

To solve 

 

     1   1      -5    6    7   -9                           

             0      1    -4    2    9

          1     -4    2    9     0

 

v   q(x)=1x³-4x²+2x+9

           r(x)=0 ;

2) Find the quotient and reminder in the division x⁵-3x³+4x-1 by x+2

To solve :

   -2   1     0     -3    0    4    -1

          0    -2     4    -2   4    -16

          1    -2     1    -2    8   -17

       

v q(x)=x⁴-2x³+x²-2x+8

r(x)= -17

3)  equation x⁴-3x³-6x²+28x-24 by x-2

 To solve :

   2    1     -3     -6     28     -24

          0     2      -2    -16     24  

          1    -1      -8      12     0

       

v    q(x)=x³-x²-8x+12

    r(x) =0  

                                                            Mysore city- India 28/03/2014