How do you solve #p^5-p>0# using a sign chart?

1 Answer
Nov 20, 2016

Answer:

The answer is #=p in ] -1,0 [uu ] 1,+oo [#

Explanation:

Let's factorise the equation

#p^5-p=p(p^4-1)=p(p^2+1)(p^2-1)#

#=p(p^2+1)(p+1)(p-1)#

The term #(p^2+1)>0#

Let #f(p)=p^5-p#

Let's do the sign chart

#color(white)(aaaa)##p##color(white)(aaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaa)##0##color(white)(aaaa)##1##color(white)(aaaa)##+oo#

#color(white)(aaaa)##p+1##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##+##color(white)(aaa)##+#

#color(white)(aaaa)##p##color(white)(aaaaaaaa)##-##color(white)(aaaa)##-##color(white)(aaa)##+##color(white)(aaa)##+#

#color(white)(aaaa)##p-1##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaa)##-##color(white)(aaa)##+#

#color(white)(aaaa)##f(p)##color(white)(aaaaa)##-##color(white)(aaaaa)##+##color(white)(aaa)##-##color(white)(aaa)##+#

So #f(p)>0# when #p in ] -1,0 [uu ] 1,+oo [#

graph{x^5-x [-8.89, 8.89, -4.444, 4.445]}