To determine the correct polynomial whose zeros are -2, -1, and 3, we need to use the fact that if a polynomial has certain zeros, we can express it in factored form. The zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero. ### Step 1: Form the Polynomial from its Zeros Given the zeros: - \( x = -2 \) - \( x = -1 \) - \( x = 3 \) We can express the polynomial \( P(x) \) in factored form as follows: \[ P(x) = (x + 2)(x + 1)(x - 3) \] ### Step 2: Expand the Polynomial Now, we will expand this product step by step. 1. **Multiply the first two factors:** \[ (x + 2)(x + 1) = x^2 + x + 2x + 2 = x^2 + 3x + 2 \] 2. **Now multiply this result by the third factor:** \[ (x^2 + 3x + 2)(x - 3) \] We will distribute \( (x - 3) \) across \( (x^2 + 3x + 2) \): - First, multiply \( x \) by each term: \[ x \cdot x^2 = x^3 \] \[ x \cdot 3x = 3x^2 \] \[ x \cdot 2 = 2x \] - Now, multiply \( -3 \) by each term: \[ -3 \cdot x^2 = -3x^2 \] \[ -3 \cdot 3x = -9x \] \[ -3 \cdot 2 = -6 \] 3. **Combine all these results:** \[ P(x) = x^3 + 3x^2 + 2x - 3x^2 - 9x - 6 \] Combine like terms: \[ P(x) = x^3 + (3x^2 - 3x^2) + (2x - 9x) - 6 \] \[ P(x) = x^3 - 7x - 6 \] ### Step 3: Identify the Correct Option Now that we have derived the polynomial \( P(x) = x^3 - 7x - 6 \), we can compare it with the options provided: - **A. \( x^3 - 7x + 6 \)**: This is incorrect because the constant term is \( +6 \) instead of \( -6 \). - **B. \( x^3 + 7x - 6 \)**: This is incorrect because the coefficient of \( x \) is \( +7 \) instead of \( -7 \). - **C. \( x^3 + 7x + 6 \)**: This is incorrect because both the coefficient of \( x \) and the constant term are incorrect. - **D. \( x^3 - 7x - 6 \)**: This is correct as it matches our derived polynomial. ### Conclusion The correct option is **D**: \( x^3 - 7x - 6 \). ### Revision Summary - A polynomial can be formed from its zeros by using the factored form. - To find the polynomial, expand the product of the factors corresponding to the zeros. - Combine like terms carefully to derive the final polynomial. - Always check each option against the derived polynomial to identify the correct answer.