To find the remainder when the polynomial \(3x^3 + 5x^2 - 11x + 4\) is divided by \(x + 3\), we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder of that division is \(f(c)\). In our case, we need to find the value of the polynomial when \(x = -3\) (since we are dividing by \(x + 3\), which can be rewritten as \(x - (-3)\)). ### Step-by-Step Explanation 1. **Identify the Polynomial**: The polynomial we are working with is: \[ f(x) = 3x^3 + 5x^2 - 11x + 4 \] 2. **Substitute \(x = -3\)**: We will evaluate \(f(-3)\): \[ f(-3) = 3(-3)^3 + 5(-3)^2 - 11(-3) + 4 \] 3. **Calculate Each Term**: - Calculate \((-3)^3\): \[ (-3)^3 = -27 \quad \Rightarrow \quad 3(-27) = -81 \] - Calculate \((-3)^2\): \[ (-3)^2 = 9 \quad \Rightarrow \quad 5(9) = 45 \] - Calculate \(-11(-3)\): \[ -11(-3) = 33 \] - The constant term is \(4\). 4. **Combine All Terms**: Now, we combine all the calculated values: \[ f(-3) = -81 + 45 + 33 + 4 \] 5. **Perform the Addition**: - First, combine \(-81\) and \(45\): \[ -81 + 45 = -36 \] - Next, add \(33\): \[ -36 + 33 = -3 \] - Finally, add \(4\): \[ -3 + 4 = 1 \] 6. **Final Result**: Thus, the remainder when \(3x^3 + 5x^2 - 11x + 4\) is divided by \(x + 3\) is: \[ \text{Remainder} = 1 \] ### Conclusion The correct option is **C. 1**. ### Explanation of Other Options: - **A. -4**: This option is incorrect because it does not match the calculated remainder. The calculations show that the remainder is positive, not negative. - **B. 4**: This option is also incorrect. While it is a positive number, it does not reflect the actual remainder we calculated. - **D. -1**: This option is incorrect as well. The calculations clearly show that the remainder is positive and not negative. ### Revision Summary: - Use the Remainder Theorem to find the remainder of a polynomial divided by a linear factor. - Substitute the value of \(x\) that makes the divisor zero into the polynomial. - Carefully calculate each term and combine them to find the final remainder. - Always double-check calculations to avoid common pitfalls in arithmetic.