Questions: f(x)=x^5-7x^4-13x^3-37x^2-68x-36

Solution Steps

To find the roots of the polynomial \( f(x) = x^5 - 7x^4 - 13x^3 - 37x^2 - 68x - 36 \), we can use numerical methods such as the numpy.roots function in Python, which computes the roots of a polynomial with given coefficients.

The polynomial \( f(x) = x^5 - 7x^4 - 13x^3 - 37x^2 - 68x - 36 \) has been analyzed, and its roots have been calculated. The roots are given as:

\[ \begin{align_} r_1 & = 9.0000 \\ r_2 & = 1.2829 \times 10^{-16} + 2i \\ r_3 & = 1.2829 \times 10^{-16} - 2i \\ r_4 & = -1.0000 \\ r_5 & = -1.0000 \end{align_} \]

From the calculated roots, we can classify them as follows:

• \( r_1 = 9.0000 \) is a real root.
• \( r_2 \) and \( r_3 \) are complex conjugate roots.
• \( r_4 \) and \( r_5 \) are repeated real roots, both equal to \(-1.0000\).

Final Answer