Question
Sun March 29, 2015 By: Dev Sultanpur123 Singh

A polynomial f(x) with rational coefficients leaves remainder 15, when divided by x – 3, 
and remainder 2x + 1, when divided by (x - 1)^2.Find the remainder when f(x) is divided by by  (x - 3) . {(x - 1)^2}
 
 

Expert Reply
Vimala Ramamurthy
Mon March 30, 2015
G i v e n space t h a t space f open parentheses x close parentheses equals p open parentheses x close parentheses open parentheses x minus 3 close parentheses plus 15 space a n d space f open parentheses x close parentheses equals q open parentheses x close parentheses open parentheses x minus 1 close parentheses squared plus 2 x plus 1 W e space n e e d space t o space f i n d space t h e space r e m a i n d e r space w h e n space f open parentheses x close parentheses space i s space d i v i d e d space b y space open parentheses x minus 3 close parentheses times open parentheses x minus 1 close parentheses squared. T h u s comma space f open parentheses x close parentheses equals s open parentheses x close parentheses times space open parentheses x minus 3 close parentheses times open parentheses x minus 1 close parentheses squared plus a x plus b... left parenthesis 1 right parenthesis B y space R e m a i n d e r space T h e o r e m comma space f open parentheses 3 close parentheses equals 15 space o r space f open parentheses 1 close parentheses equals 2 x plus 1 equals 3 S u b s t i t u t i n g space x equals 3 space a n d space x equals 1 space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma f open parentheses x close parentheses equals s open parentheses x close parentheses times space open parentheses x minus 3 close parentheses times open parentheses x minus 1 close parentheses squared plus a x plus b rightwards double arrow f open parentheses 1 close parentheses equals a open parentheses 1 close parentheses plus b rightwards double arrow 3 equals a plus b... left parenthesis 2 right parenthesis a n d f open parentheses 3 close parentheses equals a open parentheses 3 close parentheses plus b rightwards double arrow 15 equals 3 a plus b... left parenthesis 3 right parenthesis S o l v i n g space e q u a t i o n s space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma a equals 6 space a n d space b equals negative 3 T h u s comma space t h e space r e m a i n d e r space w h e n space f open parentheses x close parentheses space i s space d i v i d e d space b y space open parentheses x minus 3 close parentheses times open parentheses x minus 1 close parentheses squared space i s space 6 x minus 3.
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