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Answer :
To find the complex zeros of the polynomial function [tex]\( f(x) = x^5 - 9x^4 + 29x^3 - 45x^2 + 54x - 54 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex].
When dealing with polynomials, one effective method is to identify any obvious real roots, and then use them to factor the polynomial. However, for simplicity and clarity, let’s directly consider the result which states that the complex zeros of the polynomial are:
1. [tex]\( x = 3 \)[/tex]
2. [tex]\( x = -\sqrt{2}i \)[/tex]
3. [tex]\( x = \sqrt{2}i \)[/tex]
These answers represent the roots of the polynomial where:
- [tex]\( x = 3 \)[/tex] is a real root of the polynomial.
- [tex]\( x = -\sqrt{2}i \)[/tex] and [tex]\( x = \sqrt{2}i \)[/tex] are complex conjugates.
Since the polynomial is of degree 5 and we have found 3 zeros, there may be additional zeros or multiplicity that wasn't included in the simplified overview or result. Upon verifying, the comprehensive set of zeros in this context, confirming multiplicity and ensuring coverage are as follows:
The complex zeros are [tex]\( 3, 3, 3, -\sqrt{2}i, \sqrt{2}i \)[/tex].
Thus, the zeros are fully accounted for by noting [tex]\( 3 \)[/tex] appears three times, indicating it is a root with multiplicity 3, and the other roots are complex conjugates [tex]\( -\sqrt{2}i \)[/tex] and [tex]\( \sqrt{2}i \)[/tex].
When dealing with polynomials, one effective method is to identify any obvious real roots, and then use them to factor the polynomial. However, for simplicity and clarity, let’s directly consider the result which states that the complex zeros of the polynomial are:
1. [tex]\( x = 3 \)[/tex]
2. [tex]\( x = -\sqrt{2}i \)[/tex]
3. [tex]\( x = \sqrt{2}i \)[/tex]
These answers represent the roots of the polynomial where:
- [tex]\( x = 3 \)[/tex] is a real root of the polynomial.
- [tex]\( x = -\sqrt{2}i \)[/tex] and [tex]\( x = \sqrt{2}i \)[/tex] are complex conjugates.
Since the polynomial is of degree 5 and we have found 3 zeros, there may be additional zeros or multiplicity that wasn't included in the simplified overview or result. Upon verifying, the comprehensive set of zeros in this context, confirming multiplicity and ensuring coverage are as follows:
The complex zeros are [tex]\( 3, 3, 3, -\sqrt{2}i, \sqrt{2}i \)[/tex].
Thus, the zeros are fully accounted for by noting [tex]\( 3 \)[/tex] appears three times, indicating it is a root with multiplicity 3, and the other roots are complex conjugates [tex]\( -\sqrt{2}i \)[/tex] and [tex]\( \sqrt{2}i \)[/tex].
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