If p(x) = 4 x cubed space minus space 12 x squared plus space 14 x space minus space 3. Find remainder when p(x) is divided by 2x-1. Also verify the result by actual division.

Asked by vatsalchoudhary41 | 29th Jun, 2014, 11:27: AM

Expert Answer:

 R e m a i n d e r space T h e o r e m L e t space p open parentheses x close parentheses space b e space a n y space p o l y n o m i a l space o f space d e g r e e space g r e a t e r space t h a n space o r space e q u a l space t o space o n e space a n d space a space b e space a n y space r e a l space n u m b e r. space I f space p left parenthesis x right parenthesis space i s space d i v i d e d space b y thin space left parenthesis x minus a right parenthesis comma space t h e n space t h e space r e m a i n d e r space i s space e q u a l space t o space p left parenthesis a right parenthesis.  p open parentheses x close parentheses equals 4 x cubed minus 12 x squared plus 14 x minus 3  W e space h a v e space t o space f i n d space r e m a i n d e r space w h e n space p left parenthesis x right parenthesis space i s space d i v i d e d space b y space 2 x minus 1 equals 2 open parentheses x minus 1 half close parentheses.  B y space r e m a i n d e r space t h e o r e m comma space r e q u i r e d space r e m a i n d e r space i s space e q u a l space t o space p open parentheses 1 half close parentheses.  p left parenthesis 1 right parenthesis equals 4 open parentheses 1 half close parentheses cubed minus 12 open parentheses 1 half close parentheses squared plus 14 open parentheses 1 half close parentheses minus 3 equals 1 half minus 3 plus 7 minus 3 equals 3 over 2  A c t u a l space d i v i s i o n

Answered by Vijaykumar Wani | 30th Jun, 2014, 11:14: AM