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Bot GPT 4-o mini To find the zeroes of the polynomial \( f(x) = x^4 - 5x^3 + 11x^2 - 25x + 30 \), we can use various methods, such as synthetic division or factoring, but let's start by testing some rational roots using the Rational Root Theorem.
### Step 1: Use the Rational Root Theorem to test for possible rational roots
The possible rational roots are factors of the constant term (30), which are \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30 \).
### Step 2: Test possible rational roots
1. **Test \( x = 2 \)**:
\[
f(2) = 2^4 - 5 \cdot 2^3 + 11 \cdot 2^2 - 25 \cdot 2 + 30
\]
\[
= 16 - 40 + 44 - 50 + 30 = 0 \quad \text{(so \( x = 2 \) is a root)}
\]
2. **Test \( x = 3 \)**:
\[
f(3) = 3^4 - 5 \cdot 3^3 + 11 \cdot 3^2 - 25 \cdot 3 + 30
\]
\[
= 81 - 135 + 99 - 75 + 30 = 0 \quad \text{(so \( x = 3 \) is a root)}
\]
Now we know \( (x - 2) \) and \( (x - 3) \) are factors of \( f(x) \).
### Step 3: Perform polynomial long division or synthetic division
We can divide \( f(x) \) by \( (x - 2)(x - 3) = (x^2 - 5x + 6) \).
#### Performing synthetic division to find the quotient:
1. Dividing by \( x - 2 \):
- Coefficients: \( 1, -5, 11, -25, 30 \)
- Using synthetic division:
\[
\begin{array}{r|rrrrr}
2 & 1 & -5 & 11 & -25 & 30 \\
& & 2 & -6 & 10 & -30 \\
\hline
& 1 & -3 & 5 & -15 & 0 \\
\end{array}
\]
- Result: \( x^3 - 3x^2 + 5x - 15 \)
2. Dividing by \( x - 3 \):
\[
\begin{array}{r|rrrr}
3 & 1 & -3 & 5 & -15 \\
& & 3 & 0 & 15 \\
\hline
& 1 & 0 & 5 & 0 \\
\end{array}
\]
- Result: \( x^2 + 5 \)
Now we have:
\[
f(x) = (x - 2)(x - 3)(x^2 + 5)
\]
### Step 4: Find the roots from \( x^2 + 5 = 0 \)
Setting \( x^2 + 5 = 0 \) gives:
\[
x^2 = -5 \quad \Rightarrow \quad x = \pm \sqrt{5}i
\]
### Final Result
The roots of the polynomial \( f(x) \) are:
- \( x = 2 \)
- \( x = 3 \)
- \( x = \pm \sqrt{5} i \)
### Conclusion
Looking at the provided options, the correct choice for the roots of the polynomial is:
**2, 3, plus or minus \(\sqrt{5} i\)**