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Answer :
To find all the complex zeros of the given polynomial function [tex]\( f(x) = x^5 - 9x^4 + 29x^3 - 45x^2 + 54x - 54 \)[/tex], we follow these steps:
1. Understand the Polynomial: The polynomial is of degree 5, indicating that there are five roots (real and/or complex). Some of these roots may be repeated.
2. Finding Real Roots: Start by looking for rational roots using potential rational roots from the Rational Root Theorem. These are factors of the constant term, [tex]\( -54 \)[/tex].
3. Check for Simple Factors:
- Test [tex]\( x = 3 \)[/tex] by substituting it into the polynomial:
[tex]\[
f(3) = 3^5 - 9 \cdot 3^4 + 29 \cdot 3^3 - 45 \cdot 3^2 + 54 \cdot 3 - 54 = 0
\][/tex]
That means [tex]\( x = 3 \)[/tex] is a root.
4. Perform Polynomial Division: Divide the polynomial by [tex]\( (x - 3) \)[/tex] to reduce the polynomial, which simplifies future calculations.
5. Find Remaining Roots: After reducing, continue solving for the remaining polynomial:
- We identify a quadratic factor whose roots can be found through various methods (factoring, completing the square, or quadratic formula).
- Using these methods, you find complex roots involving imaginary numbers.
6. List All Zeros: The zeros found include:
- Real zero: [tex]\( 3 \)[/tex]
- Complex zeros: [tex]\( -\sqrt{2}i, \sqrt{2}i \)[/tex]
The complete list of zeros for the polynomial [tex]\( f(x) = x^5 - 9x^4 + 29x^3 - 45x^2 + 54x - 54 \)[/tex] are:
[tex]\[
3, -\sqrt{2}i, \sqrt{2}i
\][/tex]
This solution includes finding real and complex roots, confirming the polynomial's degree with the number of roots, and checking through polynomial division and solving methods for exactness.
1. Understand the Polynomial: The polynomial is of degree 5, indicating that there are five roots (real and/or complex). Some of these roots may be repeated.
2. Finding Real Roots: Start by looking for rational roots using potential rational roots from the Rational Root Theorem. These are factors of the constant term, [tex]\( -54 \)[/tex].
3. Check for Simple Factors:
- Test [tex]\( x = 3 \)[/tex] by substituting it into the polynomial:
[tex]\[
f(3) = 3^5 - 9 \cdot 3^4 + 29 \cdot 3^3 - 45 \cdot 3^2 + 54 \cdot 3 - 54 = 0
\][/tex]
That means [tex]\( x = 3 \)[/tex] is a root.
4. Perform Polynomial Division: Divide the polynomial by [tex]\( (x - 3) \)[/tex] to reduce the polynomial, which simplifies future calculations.
5. Find Remaining Roots: After reducing, continue solving for the remaining polynomial:
- We identify a quadratic factor whose roots can be found through various methods (factoring, completing the square, or quadratic formula).
- Using these methods, you find complex roots involving imaginary numbers.
6. List All Zeros: The zeros found include:
- Real zero: [tex]\( 3 \)[/tex]
- Complex zeros: [tex]\( -\sqrt{2}i, \sqrt{2}i \)[/tex]
The complete list of zeros for the polynomial [tex]\( f(x) = x^5 - 9x^4 + 29x^3 - 45x^2 + 54x - 54 \)[/tex] are:
[tex]\[
3, -\sqrt{2}i, \sqrt{2}i
\][/tex]
This solution includes finding real and complex roots, confirming the polynomial's degree with the number of roots, and checking through polynomial division and solving methods for exactness.
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