Solve the polynomial equation in the complex numbers.
[tex]12 x^4+32 x^3-x^2-7 x-1=0[/tex]
Find the complex zeros of [tex]f[/tex].
[tex]x=[/tex]
(Type an exact answer, using radicals as needed. Express complex numbers in terms of [tex]i[/tex]. Use a comma to separate answers as needed.)
Answer (2)
Use a polynomial root finder to approximate the roots of the equation.
Identify the exact values of the roots.
The roots are 2 − 1 − 5 , 2 1 , 2 − 1 + 5 , − 6 1 .
The complex zeros of f are x = − 6 1 , 2 1 , 2 − 1 − 5 , 2 − 1 + 5 .
Explanation
Problem Analysis We are given the polynomial equation 12 x 4 + 32 x 3 − x 2 − 7 x − 1 = 0 and asked to find its complex roots. Since the polynomial is of degree 4, it has 4 roots (counting multiplicity).
Rational Root Theorem We can attempt to find rational roots using the Rational Root Theorem. The possible rational roots are ± 1 , ± 2 1 , ± 3 1 , ± 4 1 , ± 6 1 , ± 12 1 . By testing these possible roots, we find that x = 6 − 1 and x = 2 1 are not roots.
Finding the Roots Using a polynomial root finder, we find the roots to be approximately -2.61803399, 0.5, -0.38196601, and -0.16666667. We can express these roots as − 2.61803399 , 0.5 , − 0.38196601 , − 6 1 .
Exact Roots Notice that 0.5 = 2 1 and − 6 1 = − 0.16666667 . Also, − 2.61803399 ≈ − 2 1 + 5 and − 0.38196601 ≈ 2 1 − 5 . Thus, the roots are 2 − 1 − 5 , 2 1 , 2 − 1 + 5 , − 6 1 .
Examples
Polynomial equations are used in various fields such as physics, engineering, and economics to model complex systems. For example, in physics, polynomial equations can describe the trajectory of a projectile or the behavior of electrical circuits. In economics, they can be used to model cost functions or revenue functions. Solving polynomial equations allows us to understand and predict the behavior of these systems.
The complex zeros of the polynomial equation 12 x 4 + 32 x 3 − x 2 − 7 x − 1 = 0 are x = − 6 1 , 2 1 , 2 − 1 − 5 , 2 − 1 + 5 .
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