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### Given:
- A polynomial \( f(x) = ax^3 - bx^2 + 3 \), where \( a \) and \( b \) are nonzero, real constants.
- The function has:
  - an x-intercept at \( (-5.94, 0) \)
  - a root at \( -0.27 - 0.07i \)

### Question:
Which of the following statements is true?

### Options:
1. The graph of the function \( f \) has a total of three x-intercepts.
2. The function \( f \) has three real solutions.
3. A zero of \( f \) is \( 0.27 + 0.07i \).
4. A zero of \( f \) is \( -0.27 + 0.07i \).

### Explanation:
The polynomial is cubic (\( x^3 \) term present), so it can have up to three roots (real or complex). The key details are:
- One real root is given at \( (-5.94, 0) \), which means one real x-intercept.
- A complex root \( -0.27 - 0.07i \) is also given. For polynomials with real coefficients, complex roots occur in conjugate pairs. Therefore, the root \( -0.27 - 0.07i \) implies the presence of its conjugate pair, \( -0.27 + 0.07i \).

#### Option 1:
- Incorrect. Since there are two complex roots (one conjugate pair) and one real root, there will be only **one x-intercept**, not three.

#### Option 2:
- Incorrect. The function does not have three real solutions because two of the roots are complex.

#### Option 3:
- Incorrect. The complex root is \( -0.27 - 0.07i \), not \( 0.27 + 0.07i \).

#### Option 4:
- Correct. The conjugate root to \( -0.27 - 0.07i \) is \( -0.27 + 0.07i \), so this is true.

### Final Answer:
The correct statement is **Option 4: A zero of \( f \) is \( -0.27 + 0.07i \)**.

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Do you have any questions or would you like a deeper explanation?

Here are 5 related questions you can explore:
1. How do you determine the number of real roots in a cubic function?
2. Why do complex roots always come in conjugate pairs in polynomials with real coefficients?
3. How can you graph a cubic function with both real and complex roots?
4. What role does the discriminant play in finding the nature of the roots of a cubic equation?
5. How do the coefficients \( a \) and \( b \) in the polynomial affect its shape and roots?

**Tip:** Always check the conjugate pair property when dealing with complex roots in polynomials with real coefficients.

A polynomial f(x) = ax^3 - bx^2 + 3, where a and b are nonzero, real constants. The function has an x-intercept at (-5.94, 0) and a root at -0.27 - 0.07i. Which of the following statements is true?

Analyze a cubic function with real and complex roots. Given an x-intercept at (-5.94, 0) and a root at -0.27 - 0.07i, this problem explores the graph of the function and applies the conjugate root theorem. Suitable for students in Grades 10-12.

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