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Express f(x) in the form f(x) =(x-k)q(x)+r for the given value of k.

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1 Answer

 f( X ) = 4 X^4 - 2 X^3 - 12 X^2 - X     k =4
 
        __4X ^3- 2 X^2 + 44 X + 175__________                   
 X-4  l 4 X^4 - 2X^3 - 12X ^2 -X
          4 X^4 +16 X^3             
                     14 X^3 -12X^2
                     14 X^3 +56X^2       
                                  44 X^2 - X
                                  44 X^2 + 176X
                                              175 X 
                                               175 X + 700
                                                         +700
   f (X)  =   4 X^4 - 2 X^3 - 12 X^2 - X = ( 4X^3 -2X^2 + 44 X + 175) ( X -4 ) + 700  ( 1)
           f( 4 )  = 700
    
      f ( 4) = 4 ( 4 ^4) - 2 ( 4 ^3)  - 12 ( 4 ^2 ) - 4 =
               =  4 ( 256 ) - 2 ( 64) - 12 ( 16 ) -4 = 
               = 1024 - 128 - 192  -4 = 700
 
 Note : equation (1 ) is same as dividing 17 by 4
 
            4                       
      4 l 17
           16  
            1
       17 = 4 *4 +1
 
      17 / 4 = 4 1/4       / Conversion of improper fraction to mixed number.