find the roots h(x)=27x$^{6}$ + 27x$^{3}$ - 1

asked by guest
on Sep 20, 2024 at 3:36 am



You asked:

Find the zeros of \(h\left( x \right) = 27 \cdot {x}^{6} + 27 \cdot {x}^{3} - 1\)

MathBot Answer:

The real zeros of \(h(x)=27 x^{6} + 27 x^{3} - 1\) are: $$\begin{align*}x &= \frac{\sqrt[3]{12} \sqrt[3]{-9 + \sqrt{93}}}{6} \approx 0.329452338049299\\x &= - \frac{\sqrt[3]{12} \sqrt[3]{9 + \sqrt{93}}}{6} \approx -1.01178014187732\end{align*}$$

The complex zeros of \(h(x)=27 x^{6} + 27 x^{3} - 1\) are: $$\begin{align*}x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[3]{3} \sqrt[3]{-9 + \sqrt{93}}}{12} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} i \sqrt[3]{-9 + \sqrt{93}}}{12} \approx -0.164726169024649 - 0.285314094086871 i\\x &= - \frac{2^{\frac{2}{3}} \cdot \sqrt[3]{3} \sqrt[3]{-9 + \sqrt{93}}}{12} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} i \sqrt[3]{-9 + \sqrt{93}}}{12} \approx -0.164726169024649 + 0.285314094086871 i\\x &= \frac{2^{\frac{2}{3}} \cdot \sqrt[3]{3} \sqrt[3]{9 + \sqrt{93}}}{12} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} i \sqrt[3]{9 + \sqrt{93}}}{12} \approx 0.505890070938659 - 0.876227305910381 i\\x &= \frac{2^{\frac{2}{3}} \cdot \sqrt[3]{3} \sqrt[3]{9 + \sqrt{93}}}{12} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} i \sqrt[3]{9 + \sqrt{93}}}{12} \approx 0.505890070938659 + 0.876227305910381 i\end{align*}$$

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