is it a polynomial? p(x)=2x^2 +4x^x −10


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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

	$b u t i n s o m e e x p r e s s i o n t h e y$ $c o n s i d e r e d [t h e m a s p o l y n o m e$ $s o p o l y n o m e [i s o n l y f o r$ $e x p o s a n t s w i t h t e r m e s [d i g i t s (0, + 1, + 2, + 3, . . . . .)$ $t h a n k s f o r d i s t i n c t i o n .$
	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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