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f(x) = 0 is a polynomial whose coefficients are all +-1 (plus minus 1) and whose roots are all real then the degree of f(x) can be  a) 1.  b)2.  c)3.   d)4. (Please provide a detailed solution , as we can directly linear and quadratic is possible but how can we prove that 3rd degree or 4th degree polynomial will/will not have all roots real?)
Asked by kunalgaurav39 | 23 Apr, 2019, 10:17: PM
answered-by-expert Expert Answer
C a s e space 1 colon I f space a l l space t h e space c o e f f i c i e n t s space o f space t h e space p o l y n o m i a l space f left parenthesis x right parenthesis equals 0 space a r e space 1 comma space t h e n x plus 1 equals 0 space h a s space r e a l space r o o t space w h i c h space i s space minus 1 T h e space s e c o n d space d e g r e e space p o l y n o m i a l space i. e. space x squared plus x plus 1 equals 0 space d o e s space n o t space h a v e space r e a l space r o o t s space sin c e space x equals fraction numerator negative 1 plus-or-minus i square root of 3 over denominator 2 end fraction T h e space t h i r d space d e g r e e space p o l y n o m i a l space i. e. space x cubed plus x squared plus x plus 1 equals 0 space rightwards double arrow x squared left parenthesis x plus 1 right parenthesis plus left parenthesis x plus 1 right parenthesis equals 0 rightwards double arrow left parenthesis x squared plus 1 right parenthesis left parenthesis x plus 1 right parenthesis equals 0 rightwards double arrow x equals negative 1 comma plus-or-minus i T h u s comma space a l l space t h e space r o o t s space a r e space n o t space r e a l. T h e space f o u r t h space d e g r e e space p o l y n o m i a l space a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0 space i. e. space x to the power of 4 plus x cubed plus x squared plus x plus 1 equals 0 N o w comma space d i s c r i m i n a n t space o f space a b o v e space p o l y n o m i a l space i s increment equals 256 a cubed e cubed minus 192 a squared b d e squared minus 128 a squared c squared e squared plus 144 a squared c d squared e minus 27 a squared d to the power of 4 plus 144 a b squared c e squared space space space space space space space minus 6 a b squared d squared e minus 80 a b c squared d e plus 18 a b c d cubed plus 16 a c to the power of 4 e minus 4 a c cubed d squared minus 27 b to the power of 4 e squared plus 18 b cubed c d e minus 4 b cubed d cubed minus 4 b squared c cubed e plus b squared c squared d squared H e r e comma space a equals b equals c equals d equals e equals 1 rightwards double arrow increment greater than 0 W h e r e a s space P equals 8 a c minus 3 b squared equals 8 minus 3 equals 5 greater than 0 T h u s comma space t h e space r o o t s space t w o space p a i r s space o f space n o n minus r e a l space c o m p l e x space c o n j u g a t e s S o comma space t h e space r o o t s space a r e space n o t space r e a l. C a s e space 2 colon I f space a l l space t h e space c o e f f i c i e n t s space o f space t h e space p o l y n o m i a l space f left parenthesis x right parenthesis equals 0 space a r e space minus 1 comma space t h e n minus x minus 1 equals 0 space h a s space r e a l space r o o t space w h i c h space i s space minus 1 T h e space s e c o n d space d e g r e e space p o l y n o m i a l space i. e. space minus x squared minus x minus 1 equals 0 space rightwards double arrow space x squared plus x plus 1 equals 0 space d o e s space n o t space h a v e space r e a l space r o o t s space sin c e space x equals fraction numerator negative 1 plus-or-minus i square root of 3 over denominator 2 end fraction T h e space t h i r d space d e g r e e space p o l y n o m i a l space i. e. space minus x cubed minus x squared minus x minus 1 equals 0 space rightwards double arrow x squared left parenthesis x plus 1 right parenthesis plus left parenthesis x plus 1 right parenthesis equals 0 rightwards double arrow left parenthesis x squared plus 1 right parenthesis left parenthesis x plus 1 right parenthesis equals 0 rightwards double arrow x equals negative 1 comma plus-or-minus i T h u s comma space a l l space t h e space r o o t s space a r e space n o t space r e a l. T h e space f o u r t h space d e g r e e space p o l y n o m i a l space a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0 space i. e. space minus x to the power of 4 minus x cubed minus x squared minus x minus 1 equals 0 N o w comma space d i s c r i m i n a n t space o f space a b o v e space p o l y n o m i a l space i s increment equals 256 a cubed e cubed minus 192 a squared b d e squared minus 128 a squared c squared e squared plus 144 a squared c d squared e minus 27 a squared d to the power of 4 plus 144 a b squared c e squared space space space space space space space minus 6 a b squared d squared e minus 80 a b c squared d e plus 18 a b c d cubed plus 16 a c to the power of 4 e minus 4 a c cubed d squared minus 27 b to the power of 4 e squared plus 18 b cubed c d e minus 4 b cubed d cubed minus 4 b squared c cubed e plus b squared c squared d squared H e r e comma space a equals b equals c equals d equals e equals negative 1 rightwards double arrow increment greater than 0 W h e r e a s space P equals 8 a c minus 3 b squared equals 8 minus 3 equals 5 greater than 0 T h u s comma space t h e space r o o t s space t w o space p a i r s space o f space n o n minus r e a l space c o m p l e x space c o n j u g a t e s S o comma space t h e space r o o t s space a r e space n o t space r e a l. H e n c e comma space o n l y space a space p o l y n o m i a l space o f space d e g r e e space 1 space w i l l space h a v e space r e a l space r o o t s.
Answered by Renu Varma | 25 Apr, 2019, 12:45: PM

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