How can this type of question be solved without graph . Option D is correct.If it solved using graph it is told that count the number of times it crocess the x axis,why?Can you give some facts like this to interpret graphs.

Asked by Vineeth K | 31st May, 2016, 03:14: PM

Expert Answer:

C o n s i d e r space t h e space p o l y n o m i a l comma space 2 x to the power of 4 minus 5 x cubed plus x squared minus 6 x plus 9 equals 0 L e t space f open parentheses x close parentheses equals 2 x to the power of 4 minus 5 x cubed plus x squared minus 6 x plus 9 A s s u m e space t h a t space x equals 1 comma space t h e n space f open parentheses 1 close parentheses equals 2 open parentheses 1 close parentheses to the power of 4 minus 5 open parentheses 1 close parentheses cubed plus open parentheses 1 close parentheses squared minus 6 open parentheses 1 close parentheses plus 9 equals 2 minus 5 plus 1 minus 6 plus 9 equals 1 equals plus v e A s s u m e space t h a t space x equals 2 comma space t h e n space space f open parentheses 2 close parentheses equals 2 open parentheses 2 close parentheses to the power of 4 minus 5 open parentheses 2 close parentheses cubed plus open parentheses 2 close parentheses squared minus 6 open parentheses 2 close parentheses plus 9 equals 32 minus 40 plus 4 minus 12 plus 9 equals negative 7 equals negative v e space S i n c e space t h e r e space i s space a space c h a n g e space o f space s i g n space b e t w e e n space x equals 1 space a n d space x equals 2 comma space t h e r e space s h o u l d space b e space o n e space r e a l space r o o t space b e t w e e n 1 space a n d space 2 space s u c h space t h a t space f left parenthesis x right parenthesis equals 0 S i n c e space f open parentheses x close parentheses space i s space a space p o l y n o m i a l space o f space d e g r e e space 4 comma space t h e r e space a r e space 4 space e x a c t l y space 4 space z e r o e s. W e space a l s o space k n o w space t h a t space i m a g i n a r y space r o o t s space o c c u r space i n space p a i r s. S i n c e space o n e space r o o t space i s space r e a l comma space e i t h e r space o n e space m o r e space r o o t space s h o u l d space b e space r e a l space o r space a l l space t h e space f o u r space r o o t s space o f space t h e space p o l y n o m i a l space a r e space r e a l. A s s u m e space t h a t space x equals 2 comma space t h e n space f open parentheses 2 close parentheses equals 2 open parentheses 2 close parentheses to the power of 4 minus 5 open parentheses 2 close parentheses cubed plus open parentheses 2 close parentheses squared minus 6 open parentheses 2 close parentheses plus 9 equals 32 minus 40 plus 4 minus 12 plus 9 equals negative 7 equals negative v e A s s u m e space t h a t space x equals 3 comma space t h e n space space f open parentheses 3 close parentheses equals 2 open parentheses 3 close parentheses to the power of 4 minus 5 open parentheses 3 close parentheses cubed plus open parentheses 3 close parentheses squared minus 6 open parentheses 2 close parentheses plus 9 equals 162 minus 135 plus 9 minus 18 plus 9 equals 27 equals plus v e space S i n c e space t h e r e space i s space a space c h a n g e space o f space s i g n space b e t w e e n space x equals 2 space a n d space x equals 3 comma space t h e r e space s h o u l d space b e space o n e space r e a l space r o o t space b e t w e e n 2 space a n d space 3 space s u c h space t h a t space f left parenthesis x right parenthesis equals 0 T h u s comma space t w o space r o o t s space o f space t h e space g i v e n space p o l y n o m i a l space a r e space r e a l. space a n d space w e space n e e d space t o space c h e c k space f o r space o t h e r space t w o space r o o t s A n d space i f space w e space a s s u m e space x equals a n y space n e g a t i v e space r o o t comma space f open parentheses x close parentheses space i s space a l w a y s space p o s i t i v e space a n d space t h e r e space i s space n o space c h a n g e space o f space s i g n. N o w space l e t space u s c o n s i d e r space t h e space g i v e n space f u n c t i o n space f open parentheses x close parentheses equals 2 x to the power of 4 minus 5 x cubed plus x squared minus 6 x plus 9 R e w r i t i n g space t h e space a b o v e space f u n c t i o n comma space w e space h a v e comma space f open parentheses x close parentheses equals x cubed open parentheses 2 x minus 5 close parentheses plus open parentheses x minus 3 close parentheses squared I f space x greater than 2.5 comma space f open parentheses x close parentheses greater than 0 T h u s comma space t h e r e space a r e space t w o space r e a l space r o o t s space a n d space t w o space i m a g i n a r y space r o o t s. H e n c e space c o r r e c t space o p t i o n space i s space D
O b s e r v e space t h e space g r a p h space o f space t h e space g i v e n space p o l y n o m i a l. I t space i s space c l e a r space t h a t space t h e space p o l y n o m i a l space c r o s s e s space t h e space x minus a x i s space t w i c e. T h e space e q u a t i o n space o f space x minus a x i s space i s space y equals 0 T h e r e f o r e comma space t h e space r o o t s space o f space t h e space g i v e n space p o l y n o m i a l space a r e space t h e space p o i n t s space w h e r e space t h e space g r a p h space m e e t s space t h e space x minus a x i s. H e r e comma space i t space c r o s s e s space t h e space x minus a x i s space b e t w e e n space 1 space a n d space 2 space a n d space b e t w e e n space 2 space a n d space 3 T h e r e f o r e space t h e r e space a r e space o n l y space t w o space r e a l space r o o t s

Answered by Vimala | 1st Jun, 2016, 03:47: PM

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