To find the rational zeros of the polynomial $f(x) = x^5 - 3x^4 + 2x^3 + 18x^2 - 99x + 81$, we can use the Rational Root Theorem.

Rational Root Theorem:

The possible rational roots (zeros) of the polynomial are of the form: $\text{Possible rational roots} = \frac{\text{factors of the constant term}}{\text{factors of the leading coefficient}}$ For $f(x) = x^5 - 3x^4 + 2x^3 + 18x^2 - 99x + 81$:

• The constant term is 81, so the factors of 81 are: $\pm 1, \pm 3, \pm 9, \pm 27, \pm 81$
• The leading coefficient is 1, so the factors of 1 are: $\pm 1$

Thus, the possible rational roots are: $\pm 1, \pm 3, \pm 9, \pm 27, \pm 81$

Testing the Possible Rational Roots:

We substitute each possible rational root into the polynomial $f(x)$ to check if it equals zero.

1. Testing $x = 1$: $f(1) = 1^5 - 3(1^4) + 2(1^3) + 18(1^2) - 99(1) + 81 = 1 - 3 + 2 + 18 - 99 + 81 = 0$ So, $x = 1$ is a rational zero.

2. Testing $x = -1$: $f(-1) = (-1)^5 - 3(-1)^4 + 2(-1)^3 + 18(-1)^2 - 99(-1) + 81 = -1 - 3 - 2 + 18 + 99 + 81 = 192 \neq 0$ So, $x = -1$ is not a rational zero.

3. Testing $x = 3$: $f(3) = 3^5 - 3(3^4) + 2(3^3) + 18(3^2) - 99(3) + 81 = 243 - 243 + 54 + 162 - 297 + 81 = 0$ So, $x = 3$ is a rational zero.

4. Testing $x = -3$: $f(-3) = (-3)^5 - 3(-3^4) + 2(-3^3) + 18(-3^2) - 99(-3) + 81 = -243 - 243 - 54 + 162 + 297 + 81 = 0$ So, $x = -3$ is a rational zero.

Thus, the rational zeros of the polynomial are $\boxed{1, 3, -3}$.

Final Answer:

The rational zeros are $\boxed{1, 3, -3}$.

Would you like more details or have any questions?

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Here are 5 related questions to expand on this:

1. How does the Rational Root Theorem help identify potential rational zeros?
2. What is the process to fully factor this polynomial after finding the rational zeros?
3. Can there be irrational or complex zeros in addition to the rational ones?
4. How can synthetic division be used to simplify polynomial zero-finding after identifying a root?
5. What is the difference between rational and real zeros in the context of this problem?

Tip: When testing possible rational roots, it can be faster to use synthetic division after finding a zero to reduce the polynomial degree.