Descartes' Vorzeichenregel-Rechner

Die Zeichenregel von Descartes Schritt für Schritt anwenden

Der Rechner findet die maximale Anzahl positiver und negativer reeller Wurzeln des gegebenen Polynoms unter Verwendung der Vorzeichenregel von Descartes, wobei die Schritte angezeigt werden.

Solution

Your input: find the number of real roots of $x^{3} + 7 x^{2} + 4$ using the Descartes' Rule of Signs.

The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by an even number.

So, the coefficients are $1, 7, 4$ .

As can be seen, there are $0$ changes.

This means that there are $0$ positive real roots.

To find the number of negative real roots, substitute $x$ with $- x$ in the given polynomial: $x^{3} + 7 x^{2} + 4$ becomes $- x^{3} + 7 x^{2} + 4$ .

The coefficients are $-1, 7, 4$ .

As can be seen, there is $1$ change.

This means that there is $1$ negative real root.

Answer

$0$ positive real roots.

$1$ negative real root.