Questions: Solve the polynomial equation in the complex numbers. x^3 - 4x^2 - 19x - 14 = 0 x = (Type an exact answer, using radicals as needed. Express complex numbers in terms of i. Use a comma to separate answers as needed.)

Solve the polynomial equation in the complex numbers.
x^3 - 4x^2 - 19x - 14 = 0
x =
(Type an exact answer, using radicals as needed. Express complex numbers in terms of i. Use a comma to separate answers as needed.)

Transcript text: Solve the polynomial equation in the complex numbers. \[ \begin{array}{l} x^{3}-4 x^{2}-19 x-14=0 \\ x=\square \end{array} \] (Type an exact answer, using radicals as needed. Express complex numbers in terms of $i$. Use a comma to separate answers as needed.)

Solution

Solution Steps

To solve the polynomial equation $x^3 - 4x^2 - 19x - 14 = 0$ in the complex numbers, we can use numerical methods or factorization techniques. Since the polynomial is cubic, we can attempt to find one real root using methods like the Rational Root Theorem or synthetic division. Once a real root is found, we can factor the polynomial to a quadratic and solve the remaining quadratic equation using the quadratic formula to find the other roots, which may be complex.

Step 1: Identify the Polynomial

We are given the polynomial equation:

\[ x^3 - 4x^2 - 19x - 14 = 0 \]

Step 2: Find the Roots

By solving the polynomial, we find the roots to be:

\[ x = -2, \quad x = -1, \quad x = 7 \]