Question

Hello Friends

Ques : Find the remainder of
f( x ) = 4x^3 - 12x^2 + 14x - 3, if g( x ) = 2x - 1

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⭐Find answer by Remainder Theorum⭐

Answer 1

Zero of g ( x ) = 2 x - 1 = 0

                             x = 1 / 2.

By Remainder Theorem ,

f ( x ) = 4 x^3 - 12 x^2 + 14 x - 3

By substituting value of x ,

 f ( 1/2 ) = 4 * ( 1/2 )^3 - 12 * ( 1/2 )^2 + 14 ( 1 / 2 ) - 3

 f ( 1/2 ) = 4 * ( 1/8 ) - 12 * ( 1/4 ) + 7 - 3

 f ( 1/2 ) = ( 4 * 1 ) / 8 - ( 12 * 1 ) / 4 + 7 - 3

 f ( 1/2 ) = ( 1/2 ) - 3 + 7 - 3

                   1 - 6 + 14 - 6
 f ( 1/2 ) =  --------------------------
                              2

                       15 - 12
 f ( 1/2 ) =  ------------------------
                            2

 f ( 1/2 ) = 3/2.

So, remainder = 3/2.

Answer 2

f(x) = 4x^3 - 12x^2 + 14 - 3 

g(x) = 2x  - 1


     2x - 1)4x^3 - 12x^2 + 14x - 3(2x^2 - 5x  + 9/2
              4x^3 -  2x^2
             =================
                     - 10x^2 + 14x
                     - 10x^2 + 5x
                =================
                            9x - 3
                           9x - 9/2
                       ===========
                                 (9 - 6)/2


hence the remainder is (9 - 6)/2 = 3/2


                                          AND  

By Remainder Theorem ,

f ( x ) = 4 x^3 - 12 x^2 + 14 x - 3


put the  value of x = 1/2


 f ( 1/2 ) = 4 * ( 1/2 )^3 - 12 * ( 1/2 )^2 + 14 ( 1 / 2 ) - 3
  => 4 * ( 1/8 ) - 12 * ( 1/4 ) + 7 - 3

  => ( 4 * 1 ) / 8 - ( 12 * 1 ) / 4 + 7 - 3
  => ( 1/2 ) - 3 + 7 - 3
   
  => (1 - 6 + 14 - 6) / 2

   => ( -5 + 8) / 2

   
     
    f ( 1/2 ) = 3/2.

   

 


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