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Problem: Descartes rule x^3+3x^2+3x+1 [ Calculator, Method and examples ]

Solution:
Your problem -> Descartes rule x^3+3x^2+3x+1

Here f(x)=x^3+3x^2+3x+1

f(x)=x^3color{blue}{+}3x^2color{blue}{+}3xcolor{blue}{+}1

look first at f(x): (positive case)

 f(x)= color{blue}{+} x^3 color{blue}{+} 3x^2 color{blue}{+} 3x color{blue}{+} 1 Sign change count color{blue}{+} to color{blue}{+} color{blue}{+} to color{blue}{+} color{blue}{+} to color{blue}{+}

There are no sign changes, so there are no positive roots.

Now look at f(–x): (negative case)

f(-x)=(-x)^3color{blue}{+}3(-x)^2color{blue}{+}3(-x)color{blue}{+}1

f(-x)=color{red}{-}x^3color{blue}{+}3x^2color{red}{-}3xcolor{blue}{+}1

 f(-x)= color{red}{-} x^3 color{blue}{+} 3x^2 color{red}{-} 3x color{blue}{+} 1 Sign change count color{red}{-} to color{blue}{+}1 color{blue}{+} to color{red}{-}2 color{red}{-} to color{blue}{+}3

There are 3 sign changes, so there are 3 or counting down in pairs, 1 negative roots.

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