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Let's work through this step by step.

### Part a) What is the value of \( f(1) \)?
We are given the function \( f(x) = x^3 + 7x^2 + 4x - 12 \).

To find \( f(1) \), substitute \( x = 1 \) into the function:

\[
f(1) = (1)^3 + 7(1)^2 + 4(1) - 12
\]
\[
f(1) = 1 + 7 + 4 - 12
\]
\[
f(1) = 0
\]

So, \( f(1) = 0 \).

### Part b) Using your answer to part a), select a factor of \( f(x) \).

Since \( f(1) = 0 \), this means that \( (x - 1) \) is a factor of \( f(x) \) (by the factor theorem, if \( f(a) = 0 \), then \( (x - a) \) is a factor of \( f(x) \)).

Thus, the correct factor from the options is **\( (x - 1) \)**.

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Do you want more details or have any questions on this?

Here are 5 follow-up questions to deepen your understanding:
1. How can the Factor Theorem help us identify other factors of \( f(x) \)?
2. Can we factor \( f(x) \) completely, given that \( (x - 1) \) is a factor?
3. What are other possible methods to factor cubic polynomials like \( f(x) \)?
4. How would you check if \( (x + 1) \) or \( (x - 12) \) is a factor of \( f(x) \)?
5. How would you solve \( f(x) = 0 \) for the remaining roots?

**Tip**: When dealing with polynomials, always check if substituting values into the function gives 0 to find possible factors using the Factor Theorem.

f(x) = x^3 + 7x^2 + 4x - 12
a) What is the value of f(1)?
b) Using your answer to part a), select a factor of f(x) from the options below: (x + 1), (x - 12), (x - 1), (x + 12).

Learn how to evaluate and factorize the cubic polynomial f(x) = x^3 + 7x^2 + 4x - 12. This solution demonstrates how to find the value of f(1) and use the Factor Theorem to identify a factor. Suitable for students in Grades 9-12.

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