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Solve for the roots in the equation below. In your final answer, include each of the necessary steps and calculations. Hint: Use your knowledge of polynomial division and the quadratic formula. x3 - 27 = 0

Respuesta :

[tex]\bf x^3-27=\stackrel{y}{0}\implies \stackrel{\textit{difference of cubes}}{x^3-3^3}=0\implies (x-3)(x^2+3x+3^2)=0\\\\ -------------------------------\\\\ x-3=0\implies x=3\\\\ -------------------------------\\\\ \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+3}x\stackrel{\stackrel{c}{\downarrow }}{+9}=0\implies \stackrel{\textit{using the quadratic formula}}{x=\cfrac{-3\pm\sqrt{3^2-4(1)(9)}}{2(1)}}[/tex]

[tex]\bf x={\cfrac{-3\pm\sqrt{9-36}}{2}}\implies x={\cfrac{-3\pm\sqrt{-27}}{2}}\implies x=\cfrac{-3\pm i\sqrt{27}}{2} \\\\\\ \begin{cases} 27=3\cdot 3\cdot 3\\ \qquad 3^2\cdot 3 \end{cases}\implies x=\cfrac{-3\pm i\sqrt{3^2\cdot 3}}{2}\implies x=\cfrac{-3\pm i3\sqrt{3}}{2} \\\\\\ x= \begin{cases} \frac{-3+3i\sqrt{3}}{2}\\\\ \frac{-3-3i\sqrt{3}}{2} \end{cases}[/tex]
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