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The given function is:

\[
f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12
\]

It is stated that \(x - 3\) is a factor of \(f(x)\). By the Factor Theorem, if \(x - 3\) is a factor, then \(f(3) = 0\). Let's verify that by calculating \(f(3)\).

I'll compute \(f(3)\) by substituting \(x = 3\) into the function:

\[
f(3) = 2(3)^5 + (3)^4 - 18(3)^3 - 17(3)^2 + 20(3) + 12
\]

Let me calculate this value.The value of \( f(3) \) is 0, confirming that \( x - 3 \) is indeed a factor of \( f(x) \), as expected from the Factor Theorem.

Would you like further details or have any other questions?

Here are five related questions you might find useful:
1. How does the Factor Theorem help in finding factors of a polynomial?
2. Can \( f(x) \) have other factors besides \( x - 3 \)?
3. How would you completely factor \( f(x) \)?
4. What happens if we substitute other values of \( x \) into \( f(x) \)?
5. How can we use synthetic division to confirm \( x - 3 \) is a factor?

**Tip:** When testing potential factors of a polynomial, always substitute the corresponding root into the function to verify if it results in zero.

The function f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12 is given. If x - 3 is a factor, what is the value of f(3)?

Learn how to apply the Factor Theorem to the polynomial function f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12 to find the value of f(3). This problem involves key polynomial concepts and is suitable for students in Grades 10-12.

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