Questions: Find all solutions: 3 y^3 - 5 y^2 + 34 y + 12 = 0 The solutions are y = (Enter your answers, separated by commas)

Find all solutions: 3 y^3 - 5 y^2 + 34 y + 12 = 0 The solutions are y = (Enter your answers, separated by commas)
Transcript text: Find all solutions: \[ 3 y^{3}-5 y^{2}+34 y+12=0 \] The solutions are $y=$ $\square$ (Enter your answers, separated by commas)
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Solution

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Solution Steps

To solve the cubic equation \(3y^3 - 5y^2 + 34y + 12 = 0\), we can use numerical methods to find the roots. One common approach is to use the numpy library in Python, which provides a function to find the roots of a polynomial.

Step 1: Identify the Polynomial Equation

We start with the cubic equation: \[ 3y^3 - 5y^2 + 34y + 12 = 0 \]

Step 2: Find the Roots of the Polynomial

The roots of the polynomial are: \[ y_1 = 1 + 3.3166i \] \[ y_2 = 1 - 3.3166i \] \[ y_3 = -0.3333 \]

Final Answer

\[ \boxed{y = 1 + \sqrt{11}i, 1 - \sqrt{11}i, -\frac{1}{3}} \]

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