nainder when \( x^{3}-11 x+14 \) is divided by \( x-2 \) by completing the parts

To find the remainder when the polynomial \( f(x) = x^{3} - 11x + 14 \) is divided by \( x - 2 \), we can use two methods: the **Remainder Theorem** or **Polynomial Long Division**. Below, both methods are detailed step-by-step. --- ### **Method 1: Remainder Theorem** The **Remainder Theorem** states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). 1. **Identify \( a \):** - Here, the divisor is \( x - 2 \), so \( a = 2 \). 2. **Evaluate \( f(2) \):** \[ f(2) = (2)^3 - 11(2) + 14 = 8 - 22 + 14 = 0 \] 3. **Conclusion:** - The remainder is **0**. --- ### **Method 2: Polynomial Long Division** We'll divide \( f(x) = x^{3} - 11x + 14 \) by \( x - 2 \) step-by-step. 1. **Set Up the Division:** \[ \begin{array}{r|l} x - 2 & x^{3} + 0x^{2} - 11x + 14 \\ \end{array} \] *(Note: Added \( 0x^{2} \) for clarity.)* 2. **First Division:** - **Divide:** \( x^{3} \) by \( x \) to get \( x^{2} \). - **Multiply:** \( x^{2} \times (x - 2) = x^{3} - 2x^{2} \). - **Subtract:** \[ (x^{3} + 0x^{2}) - (x^{3} - 2x^{2}) = 2x^{2} \] - **Bring Down:** \( -11x \). - **New Dividend:** \( 2x^{2} - 11x \). 3. **Second Division:** - **Divide:** \( 2x^{2} \) by \( x \) to get \( 2x \). - **Multiply:** \( 2x \times (x - 2) = 2x^{2} - 4x \). - **Subtract:** \[ (2x^{2} - 11x) - (2x^{2} - 4x) = -7x \] - **Bring Down:** \( +14 \). - **New Dividend:** \( -7x + 14 \). 4. **Third Division:** - **Divide:** \( -7x \) by \( x \) to get \( -7 \). - **Multiply:** \( -7 \times (x - 2) = -7x + 14 \). - **Subtract:** \[ (-7x + 14) - (-7x + 14) = 0 \] 5. **Conclusion:** - The **remainder is 0**. --- ### **Final Answer** The remainder is **0**.