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

Nov 20, 2016

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

#### Explanation:

Let's factorise the equation

${p}^{5} - p = p \left({p}^{4} - 1\right) = p \left({p}^{2} + 1\right) \left({p}^{2} - 1\right)$

$= p \left({p}^{2} + 1\right) \left(p + 1\right) \left(p - 1\right)$

The term $\left({p}^{2} + 1\right) > 0$

Let $f \left(p\right) = {p}^{5} - p$

Let's do the sign chart

$\textcolor{w h i t e}{a a a a}$$p$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$p + 1$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$p$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$p - 1$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(p\right)$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$

So $f \left(p\right) > 0$ when p in ] -1,0 [uu ] 1,+oo [

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