is-it-a-polynomial-p-x-2x-2-4x-x-10-

Question Number 188822 by sciencestudentW last updated on 07/Mar/23

i s i t a p o l y n o m i a l ?

p (x) = 2 x^{2} + 4 x^{x} - 10

Commented by mr W last updated on 07/Mar/23

y o u s e e m n o t t o h a v e r e a d t h e

d e f i n i t i o n o f p o l y n o m i a l .

Commented by mr W last updated on 07/Mar/23

Commented by sciencestudentW last updated on 07/Mar/23

t h a n k s s i r

Answered by a.lgnaoui last updated on 07/Mar/23

N o; p (x) i s i n f o r m a t :

({a x}^{n} + b) + x^{x} (a = n = 2)

f i r s t t e r m e 2 x^{2} + 4 x - 10 i s

p o l y n o m i n a l b u t t h e s e c o n d t e r m e

i s n o t

w u i h a p l i c a t i o n o f a d d i t i o n

p o l y n o m e s :

S u m, p r o d u i t [o f 2 o r m o r e p o l y n o m e s] i s

a p o l y n o m e

s o; E x p r e s s i o n l i k e y o u r s

s h o w t h a t i s n o t p o l y n o m e

b e c a u s e t h e (e x p o s a n t o f x^{x})

m u s t b e t e r m e \in Q)

x^{x} = 1 . x^{x} (x n o n d e f i n i)

e x p o s a n t x : m u s t b e :

= m (m \in Z) o r = (\frac{p}{q}) (w i t h p, q \in Z)

{a x}^{n} : n a m e : M o n o m e

{a x}^{n} + {b x}^{m} o r ({a x}^{n} + p) : b i n o m e

a s (b i n o m e o f N e w t o n) .

{a x}^{n} + {b x}^{m} + {c x}^{p} + d (p o y n o m e

(n b t e r m e s >= 3) (n, m, p \in Z)

Commented by sciencestudentW last updated on 07/Mar/23

{t h a t}^{'} s g r e a t t h a n k s s i r

Commented by mr W last updated on 07/Mar/23

w r o n g!

x^{\frac{p}{q}} i s n o t p o l y n o m i a l f o r p, q \in Z

2 x^{\frac{2}{3}} + 3 x^{2} + 4 i s n o t p o l y n o m i a l .

Commented by a.lgnaoui last updated on 07/Mar/23

b u t i n s o m e e x p r e s s i o n t h e y

c o n s i d e r e d [t h e m a s p o l y n o m e

s o p o l y n o m e [i s o n l y f o r

e x p o s a n t s w i t h t e r m e s [d i g i t s (0, + 1, + 2, + 3, \dots . .)

t h a n k s f o r d i s t i n c t i o n .

Commented by mr W last updated on 08/Mar/23

y e s, o n l y p o s i t i v e i n t e g e r p o w e r o f

v a r i a b l e s .

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