Let's break down the given information and use it to find the correct answer step by step.

Information Given:

1. The polynomial is $f(x) = ax^3 - bx^2 + 3$, where $a$ and $b$ are nonzero real constants.
2. The function has an x-intercept at $(-5.94, 0)$. This means that $x = -5.94$ is a real root of $f(x)$.
3. The function has a root at $-0.27 - 0.07i$, which is a complex number.

Step 1: Understanding the nature of the roots

Since $f(x)$ is a cubic polynomial (degree 3), it can have:

• 3 real roots, or
• 1 real root and 2 complex roots.

However, complex roots for polynomials with real coefficients always come in conjugate pairs. This means that if $-0.27 - 0.07i$ is a root, its conjugate $-0.27 + 0.07i$ must also be a root.

Step 2: Finding the number of real and complex roots

The function has:

• One known real root at $x = -5.94$.
• Two complex roots: $-0.27 - 0.07i$ and $-0.27 + 0.07i$.

Thus, the function has exactly one real root and two complex roots.

Step 3: Evaluating the options

• Option 1: "The graph of the function $f$ has a total of three x-intercepts."

  • Incorrect. The function has only one real root and two complex roots, which means it has only one x-intercept.

• Option 2: "The function $f$ has three real solutions."

  • Incorrect. As we determined, the function has one real root and two complex roots, not three real solutions.

• Option 3: "A zero of $f$ is $0.27 + 0.07i$."

  • Incorrect. The complex root is $-0.27 + 0.07i$, not $0.27 + 0.07i$.

• Option 4: "A zero of $f$ is $-0.27 + 0.07i$."

  • Correct! This is the conjugate pair of $-0.27 - 0.07i$, and thus a valid zero of $f(x)$.

Final Answer:

The correct statement is:

• A zero of $f$ is $-0.27 + 0.07i$.

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Would you like further details or clarification on any part of this solution?

Here are 5 related questions to consider:

1. How do complex roots affect the shape of a cubic polynomial graph?
2. What is the relationship between complex conjugates and real coefficients?
3. How do you determine the number of real vs complex roots in a cubic function?
4. What happens to the real root if the polynomial has more than one complex pair?
5. How can we factor a cubic polynomial when complex roots are involved?

Tip: For any polynomial with real coefficients, if it has a complex root, its conjugate must also be a root.