Let's analyze the information provided in the image:

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 $-0.27 + 0.07i$.

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.