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The real roots of the cubic and quartic polynomials are studied geometricallywith the help of their respective Siebeck--Marden--Northshield equilateraltriangle and regular tetrahedron. The Vi\`ete trigonometric formulae for theroots of the cubic are established through the rotation of the triangle byvariation of the free term of the cubic. A very detailed complete rootclassification for the quartic $x^4 + ax^3 + bx^2 + cx + d$ is proposed forwhich the conditions are imposed on the individual coefficients $a$, $b$, $c$,and $d$. The maximum and minimum lengths of the interval containing the fourreal roots of the quartic are determined in terms of $a$ and $b$. The upper andlower root bounds for a quartic with four real roots are also found: no rootcan lie farther than $(\sqrt{3}/4)\sqrt{3a^2 - 8b}\, $ from $-a/4$. The realroots of the quartic are localized by finding intervals containing at most tworoots. The end-points of these intervals depend on $a$ and $b$ and are roots ofquadratic equations -- which makes this localization helpful for quarticequations with complicated parametric coefficients.

On the Cubic Equation with its Siebeck--Marden--Northshield Triangle and the Quartic Equation with its Tetrahedron