\begin{tabular}{l} Questions \ Use the Fundamental Theorem of Algebra and the Complex Conjugates Theorem.) \ $$ x^{5}-9 x^{4}+14 x^{2}+15 x-13=0 $$ \ \hline yes \ \hline\end{tabular}

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To analyze the polynomial $$ f(x) = x^{5} - 9x^{4} + 14x^{2} + 15x - 13 $$ using the Fundamental Theorem of Algebra and the Complex Conjugates Theorem, we can follow these steps:

1. **Fundamental Theorem of Algebra**: This theorem states that a polynomial of degree $$ n $$ has exactly $$ n $$ roots in the complex number system, counting multiplicities. Since our polynomial is of degree 5, it will have exactly 5 roots (real or complex).

2. **Complex Conjugates Theorem**: This theorem states that if a polynomial has real coefficients, then any non-real complex roots must occur in conjugate pairs. This means if $$ a + bi $$ is a root (where $$ a $$ and $$ b $$ are real numbers and $$ b 
eq 0 $$), then its conjugate $$ a - bi $$ is also a root.

Now, let's analyze the polynomial $$ f(x) $$:

- The polynomial is of degree 5, so it has 5 roots in total.
- The coefficients of the polynomial are all real numbers.

### Finding the Roots

To find the roots, we can use numerical methods or graphing techniques, as finding roots analytically for higher-degree polynomials can be complex.

1. **Graphing**: By graphing the polynomial, we can identify any real roots.

2. **Numerical Methods**: We can also use methods like the Newton-Raphson method or synthetic division to find approximate roots.

### Conclusion

Since the polynomial is of degree 5 and has real coefficients, we can conclude:

- There are 5 roots in total.
- If there are any complex roots, they will appear in conjugate pairs.

If you need specific numerical roots or further analysis, please let me know!
The polynomial $$ x^{5}-9x^{4}+14x^{2}+15x-13 $$ has 5 roots due to the Fundamental Theorem of Algebra. Since it has real coefficients, any complex roots will come in conjugate pairs.

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Use the Fundamental Theorem of Algebra and the Complex Conjugates Theorem.)

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