Which of the following are polynomials and why
(i)(x-2)(x+3)
(ii)1 over x squared plus space 1 over x cubed minus space 3
(iii)left parenthesis x minus 1 right parenthesis left parenthesis x squared plus space x space plus space 1 right parenthesis
(iv)t cubed space minus space square root of 2 t space plus space square root of 3
(v)square root of x to the power of 5 end root plus space x squared space plus space 7 x

Asked by vatsalchoudhary41 | 29th Jun, 2014, 11:38: AM

Expert Answer:

left parenthesis i right parenthesis S i n c e space open parentheses x minus 2 close parentheses open parentheses x plus 3 close parentheses equals x squared plus 3 x minus 2 x minus 6 equals x squared plus x minus 6 comma space open parentheses x minus 2 close parentheses open parentheses x plus 3 close parentheses space i s space a space p o l y n o m i a l. left parenthesis i i right parenthesis 1 over x squared plus 1 over x cubed minus 3 equals x to the power of minus 2 end exponent plus x to the power of minus 3 end exponent minus 3 P o l y n o m i a l space i s space a n space e x p r e s s i o n space c o n s i s t i n g space o f space v a r i a b l e s space a n d space c o e f f i c i e n t s space t h a t space i n v o l v e s space o n l y space t h e space o p e r a t i o n s space o f space a d d i t i o n comma space s u b t r a c t i o n space m u l t i p l i c a t i o n space a n d space n o n minus n e g a t i v e space i n t e g e r space e x p o n e n t s comma I n space t h i s space e x a m p l e comma space v a r i a b l e space i s space h a v i n g space t h e space n e g a t i v e space i n t e g e r space e x p o n e n t s space a n d space h e n c e space i t space i s space n o t space a space p o l y n o m i a l left parenthesis i i i right parenthesis S i n c e space open parentheses x minus 1 close parentheses open parentheses x squared plus x plus 1 close parentheses equals x cubed plus x squared plus x minus x squared minus x minus 1 equals x cubed minus 1 comma space t h e space g i v e n space p r o d u c t space o f space t h e space e x p r e s s i o n s space i s space a space p o l y n o m i a l. left parenthesis i v right parenthesis space S i n c e space t cubed minus square root of 2 t plus square root of 3 space s a t i s f i e s space t h e space d e f i n i t i o n space o f space a space p o l y n o m i a l space a n d space t h e space p o w e r s space o f space t h e space v a r i a b l e space a r e space p o s i t i v e i n t e g e r space e x p o n e n t s comma space space t cubed minus square root of 2 t plus square root of 3 space space i s space a space p o l y n o m i a l. left parenthesis v right parenthesis square root of x to the power of 5 end root plus x squared plus 7 x I n space t h e space a b o v e space e x a m p l e comma space t h e space p o w e r s space o f space t h e space v a r i a b l e space a r e space n o t space i n t e g r a l space e x p o n e n t s comma space t h e space g i v e n e x p r e s s i o n space i s space n o t space a space p o l y n o m i a l.

Answered by Vimala Ramamurthy | 30th Jun, 2014, 11:00: AM