Questions: Solve. 3x^3-x^2+27x-9=0 Write imaginary solutions in terms of i. If there are multiple solutions, separate the answers with commas. x=

Solution Steps

To solve the cubic equation \(3x^3 - x^2 + 27x - 9 = 0\), we can use numerical methods or a library that handles polynomial equations. The numpy library in Python provides a convenient function to find the roots of a polynomial. We will represent the polynomial with its coefficients and use numpy.roots to find all solutions, including imaginary ones.

The roots of the polynomial \(3x^3 - x^2 + 27x - 9 = 0\) are calculated as follows:

• \(x_1 = 3.8257 \times 10^{-18} + 3i\)
• \(x_2 = 3.8257 \times 10^{-18} - 3i\)
• \(x_3 = 0.3333\)

The roots include two complex conjugate pairs and one real root:

• The complex roots are \(x_1 = 3i\) and \(x_2 = -3i\).
• The real root is \(x_3 = \frac{1}{3}\).

Final Answer