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  if alpha & beeta are zeroes of polynomial .  if (x) =x^2+px+4s (alpha-beta)^2=144 find p

Asked by seeni2005 14th May 2016, 11:35 AM
Answered by Expert
Answer:
begin mathsize 16px style C o n s i d e r space t h e space g i v e n space p o l y n o m i a l comma
f open parentheses x close parentheses equals x squared plus p x plus 45
L e t space alpha space a n d space beta space a r e space t h e space z e r o e s space o f space t h e space a b o v e space p o l y n o m i a l.
T h u s comma
alpha plus beta equals negative p space a n d space alpha cross times beta equals 45 space... left parenthesis 1 right parenthesis

W e space k n o w space t h a t comma
open parentheses alpha minus beta close parentheses squared equals alpha squared minus 2 alpha beta plus beta squared
space space space space space space space space space space space space equals alpha squared plus 2 alpha beta plus beta squared minus 4 alpha beta
space space space space space space space space space space space space equals open parentheses alpha squared plus 2 alpha beta plus beta squared close parentheses minus 4 alpha beta
space space space space space space space space space space space space equals open parentheses alpha plus beta close parentheses squared minus 4 alpha beta
T h u s comma space open parentheses alpha minus beta close parentheses squared space equals open parentheses alpha plus beta close parentheses squared minus 4 alpha beta space... left parenthesis 2 right parenthesis

S u b s t i u t i n g space t h e space v a l u e s space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis space i n space e q u a t i o n space left parenthesis 2 right parenthesis comma space w e space h a v e comma
open parentheses alpha minus beta close parentheses squared space equals open parentheses negative p close parentheses squared minus 4 cross times 45
space space space space space space space space space space space space space space equals p squared minus 180 space space space... left parenthesis 3 right parenthesis
G i v e n space t h a t space open parentheses alpha minus beta close parentheses squared equals 144... left parenthesis 4 right parenthesis
T h u s comma space e q u a t i n g space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space h a v e comma
p squared minus 180 equals 144
rightwards double arrow p squared equals 324
rightwards double arrow p equals plus-or-minus 18

N o t e colon
P l e a s e space c h e c k space t h e space q u e s t i o n
Q u e s t i o n space s h o u l d space b e colon
I f space alpha space a n d space beta space a r e space z e r o s space o f space p o l y n o m i a l comma space i f space f left parenthesis x right parenthesis equals x squared plus p x plus 45 space a n d space open parentheses alpha minus beta close parentheses squared equals 144 comma space f i n d space p.
P l e a s e space c h e c k.


end style
Answered by Expert 14th May 2016, 1:57 PM
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