To solve the equation [tex]\( x^4 + 3x^3 - 27x = 81 \)[/tex], we first write it in standard form:
[tex]\[ x^4 + 3x^3 - 27x - 81 = 0 \][/tex]
Now, we need to factor or find the roots of this polynomial equation. In this case, the equation has the following roots:
1. [tex]\( x = -3 \)[/tex]
2. [tex]\( x = 3 \)[/tex]
3. [tex]\( x = \frac{-3}{2} - \frac{3\sqrt{3}i}{2} \)[/tex]
4. [tex]\( x = \frac{-3}{2} + \frac{3\sqrt{3}i}{2} \)[/tex]
Let's verify these solutions:
1. Substituting [tex]\( x = -3 \)[/tex] into the equation:
[tex]\[
(-3)^4 + 3(-3)^3 - 27(-3) - 81 = 81 - 81 = 0
\][/tex]
This solution is correct.
2. Substituting [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[
3^4 + 3(3^3) - 27 \cdot 3 - 81 = 81 - 81 = 0
\][/tex]
This solution is correct.
3. Substituting [tex]\( x = \frac{-3}{2} - \frac{3\sqrt{3}i}{2} \)[/tex] into the equation:
Confirming through algebraic or numerical simplification shows it satisfies the polynomial equation.
4. Substituting [tex]\( x = \frac{-3}{2} + \frac{3\sqrt{3}i}{2} \)[/tex] into the equation:
Confirming through algebraic or numerical simplification shows it satisfies the polynomial equation.
Hence, the roots of the equation:
[tex]\[ x^4 + 3x^3 - 27x - 81 = 0 \][/tex]
are [tex]\(\pm 3\)[/tex] and [tex]\(\frac{-3 \pm 3i\sqrt{3}}{2}\)[/tex].
Therefore, the correct choice is:
[tex]\[ \pm 3; \frac{-3 \pm 3i \sqrt{3}}{2} \][/tex]
