# Step-by-Step Solution ## Problem Restatement **Given:** Zeros: \( 6, -6i \) **Task:** Find a polynomial function with real coefficients that has these zeros. --- ### **Step 1: Identify All Zeros** If a polynomial has real coefficients, then complex zeros always come in conjugate pairs. Given zero: \( -6i \) So, its conjugate \( 6i \) must also be a zero. **All zeros:** \( 6, -6i, 6i \) --- ### **Step 2: Write Factors** For zeros \( a, b, c \), the polynomial is: \[ (x-a)(x-b)(x-c) \] So, the factors are: \[ (x-6), (x+6i), (x-6i) \] --- ### **Step 3: Multiply the Factors** First, multiply the complex conjugates: \[ (x+6i)(x-6i) = x^2 - (6i)^2 = x^2 - (36)(-1) = x^2 + 36 \] Now, multiply by the remaining factor: \[ (x-6)(x^2 + 36) \] Expand: \[ x(x^2 + 36) - 6(x^2 + 36) = x^3 + 36x - 6x^2 - 216 \] \[ = x^3 - 6x^2 + 36x - 216 \] --- ### **Step 4: Match with the Options** Let's check the options: A) \( x^3 + 6x^2 + 36x + 216 \) B) \( x^3 - 6x^2 + 36x + 216 \) C) \( x^3 + 6x^2 - 36x + 216 \) D) \( x^3 - 6x^2 + 36x - 216 \) E) \( x^3 + 6x^2 + 36x + 216 \) Our result: \( x^3 - 6x^2 + 36x - 216 \) **So the correct answer is:** --- ## **Final Answer** **Option D: \( \boxed{x^3 - 6x^2 + 36x - 216} \)** --- ### **Explanation for Each Option** - **A & E:** Incorrect sign for \( x^2 \) term and constant. - **B:** Constant term sign is incorrect. - **C:** Incorrect sign for \( x^2 \) and \( x \) terms. - **D:** **Correct!** Matches our expanded polynomial. --- **Summary:** The polynomial with zeros \( 6, -6i, 6i \) and real coefficients is \[ \boxed{x^3 - 6x^2 + 36x - 216} \]