Evaluate: n(x)=x^5-5x^3

Expression: $n\left( x \right)={x}^{5}-5{x}^{3}$

To find the $x$-intercept/zero, substitute $n\left( x \right)=0$

$0={x}^{5}-5{x}^{3}$

Swap the sides of the equation

${x}^{5}-5{x}^{3}=0$

Factor out ${x}^{3}$ from the expression

${x}^{3} \times \left( {x}^{2}-5 \right)=0$

When the product of factors equals $0$, at least one factor is $0$

$\begin{array} { l }{x}^{3}=0,\\{x}^{2}-5=0\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=0,\\{x}^{2}-5=0\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=0,\\x=-\sqrt{ 5 },\\x=\sqrt{ 5 }\end{array}$

The equation has $3$ solutions

$\begin{align*}&\begin{array} { l }x_1=-\sqrt{ 5 },& x_2=0,& x_3=\sqrt{ 5 }\end{array} \\&\begin{array} { l }x_1\approx-2.23607,& x_2=0,& x_3\approx2.23607\end{array}\end{align*}$