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Answer :
Sure, let's go through the steps to find all the zeroes of the function [tex]\( f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45 \)[/tex].
### Step 1: State the polynomial function
The polynomial function we are working with is:
[tex]\[ f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45 \][/tex]
### Step 2: Use factorization or a root-finding method
To find the roots of the polynomial, you can use methods such as synthetic division, the Rational Root Theorem, or more advanced techniques like finding complex roots. For the sake of clarity and simplicity, we focus on the result which provides us with the roots.
### Step 3: Identify the roots of the polynomial
After applying the appropriate method to solve for the roots of the polynomial, we find the roots to be:
[tex]\[ -3, \quad 3, \quad 2 - i, \quad 2 + i \][/tex]
### Step 4: Verify the roots (optionally)
To ensure these are correct, you could substitute each root back into the original polynomial to check that [tex]\( f(x) = 0 \)[/tex]. However, the given roots are correct and there is no need to re-verify every step analytically here.
### Conclusion
Therefore, the zeroes of the function [tex]\( f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45 \)[/tex] are:
[tex]\[ \boxed{-3, \quad 3, \quad 2 - i, \quad 2 + i} \][/tex]
These roots fully explain the solutions to the given polynomial equation.
### Step 1: State the polynomial function
The polynomial function we are working with is:
[tex]\[ f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45 \][/tex]
### Step 2: Use factorization or a root-finding method
To find the roots of the polynomial, you can use methods such as synthetic division, the Rational Root Theorem, or more advanced techniques like finding complex roots. For the sake of clarity and simplicity, we focus on the result which provides us with the roots.
### Step 3: Identify the roots of the polynomial
After applying the appropriate method to solve for the roots of the polynomial, we find the roots to be:
[tex]\[ -3, \quad 3, \quad 2 - i, \quad 2 + i \][/tex]
### Step 4: Verify the roots (optionally)
To ensure these are correct, you could substitute each root back into the original polynomial to check that [tex]\( f(x) = 0 \)[/tex]. However, the given roots are correct and there is no need to re-verify every step analytically here.
### Conclusion
Therefore, the zeroes of the function [tex]\( f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45 \)[/tex] are:
[tex]\[ \boxed{-3, \quad 3, \quad 2 - i, \quad 2 + i} \][/tex]
These roots fully explain the solutions to the given polynomial equation.
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