How to prove a system is stable if the characteristic equation have non zero positive coefficients?
This is very likely a reference to Descartes' rule of signs. Seems like the statement in the original question should only apply when a polynomial has exclusively real roots. The original statement was already disproven by @jDAQ with a counter-example, where \$s^3+s^2+s+10\$ has two roots with positive real parts, even though all coefficients are positive.
Applying Descartes' rule to \$s^3+s^2+s+10\$, we should first see the number of coefficient sign changes (0) and draw the conclusion the polynomial &quot;has 0 positive real roots&quot;, which is true (it has 2 right-hand-side complex roots instead). By inverting the sign of odd-powered terms, we then see 3 sign transitions, leading to the conclusion the polynomial &quot;has 3 or 1 negative real roots&quot;. Therefore, Descartes' rule gives us two possible predictions on the roots:

\$s^3+s^2+s+10\$ has 3 negative real roots (incorrect prediction)
\$s^3+s^2+s+10\$ has 1 negative real root and two complex roots (correct prediction)

The first prediction would be confirmed for a polynomial such as \$s^3+10s^2+10s+1\$. It has the same coefficient signs as the counter-example and has 3 negative real roots (\$-1\$, \$-8.89\$ and \$-0.11\$).
In summary, the original statement from the question

I read that a control system would be stable if its characteristic equation has got all the terms present and are positive.

only applies in full to polynomials with real roots, since counter-examples with complex roots can be found, as already predicted in Descartes' rule of signs.