when a polynomial ax^3+bx^3+x-6 is a factor of (x-2) and leaves remainder 4  when divided by (x-2). find a

Asked by Hrishikesh marathe | 13th Jun, 2016, 07:31: PM

Expert Answer:

begin mathsize 14px style left parenthesis x plus 2 right parenthesis space i s space a space f a c t o r space o f space a x cubed plus b x squared plus x minus 6.
rightwards double arrow p left parenthesis negative 2 right parenthesis equals 0
rightwards double arrow a left parenthesis negative 2 right parenthesis cubed plus b left parenthesis negative 2 right parenthesis squared plus left parenthesis negative 2 right parenthesis minus 6 equals 0
rightwards double arrow negative 8 a plus 4 b minus 2 minus 6 equals 0
rightwards double arrow negative 8 a plus 4 b minus 8 equals 0
rightwards double arrow negative 2 a plus b minus 2 equals 0
rightwards double arrow 2 a minus b equals negative 2 space space space space space.... left parenthesis 1 right parenthesis
G i v e n space t h a t space w h e n space space i s space d i v i d e d space b y space left parenthesis x minus 2 right parenthesis comma space l e a v e s space r e m a i n d e r space 4.
rightwards double arrow p left parenthesis 2 right parenthesis equals 4
rightwards double arrow a left parenthesis 2 right parenthesis cubed plus b left parenthesis 2 right parenthesis squared plus 2 minus 6 equals 4
rightwards double arrow 8 a plus 4 b minus 4 equals 4
rightwards double arrow 8 a plus 4 b equals 8
rightwards double arrow 2 a plus b equals 2 space space space space space space space.... left parenthesis 2 right parenthesis
A d d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e
4 a equals 0 rightwards double arrow a equals 0
T h e n comma space 2 left parenthesis 0 right parenthesis minus b equals negative 2
rightwards double arrow 0 minus b equals negative 2
rightwards double arrow b equals 2
space end style

Answered by Rashmi Khot | 14th Jun, 2016, 09:07: AM

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