Questions: f(x)=4x^4-6x^2-3x-1

Transcript text: b. $f(x)=4 x^{4}-6 x^{2}-3 x-1$

Solution

Solution Steps

To find the roots of the polynomial function $ f(x) = 4x^4 - 6x^2 - 3x - 1 $, we can use numerical methods since it is a quartic equation. One common approach is to use Python's numpy library, which provides a function to find the roots of a polynomial.

Step 1: Identify the Roots

The polynomial $ f(x) = 4x^4 - 6x^2 - 3x - 1 $ has been analyzed, and its roots have been determined to be:

\[ \begin{align_} r_1 & = 1.4598 \\ r_2 & = -1.0 \\ r_3 & = -0.2299 + 0.3441i \\ r_4 & = -0.2299 - 0.3441i \end{align_} \]

Step 2: Classify the Roots

The roots can be classified as follows:

$ r_1 = 1.4598 $ is a real root.
$ r_2 = -1.0 $ is another real root.
$ r_3 = -0.2299 + 0.3441i $ and $ r_4 = -0.2299 - 0.3441i $ are complex conjugate roots.

Final Answer

The roots of the polynomial $ f(x) = 4x^4 - 6x^2 - 3x - 1 $ are:

\[ \boxed{1.4598, -1.0, -0.2299 + 0.3441i, -0.2299 - 0.3441i} \]

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