Determine whether the given binomial is a factor of the polynomial \( p(x) \). If it is, find the remaining factors of \( p(x) \). 2. \( p(x)=x^{3}-2 x^{2}+7 x-14 ; x-2 \)

To determine whether \( x - 2 \) is a factor of the polynomial \( p(x) = x^{3} - 2x^{2} + 7x - 14 \), and to find the remaining factors if it is, follow these steps: ### Step 1: Check if \( x - 2 \) is a Factor Use the **Factor Theorem**, which states that \( x - a \) is a factor of \( p(x) \) if and only if \( p(a) = 0 \). Evaluate \( p(2) \): \[ p(2) = (2)^3 - 2(2)^2 + 7(2) - 14 = 8 - 8 + 14 - 14 = 0 \] Since \( p(2) = 0 \), \( x - 2 \) is indeed a factor of \( p(x) \). ### Step 2: Divide \( p(x) \) by \( x - 2 \) to Find the Remaining Factor Perform polynomial division (synthetic division is efficient here): **Synthetic Division Setup:** \[ \begin{array}{c|cccc} 2 & 1 & -2 & 7 & -14 \\ \hline & & 2 & 0 & 14 \\ \hline & 1 & 0 & 7 & 0 \\ \end{array} \] **Explanation:** 1. Bring down the leading coefficient: **1**. 2. Multiply by 2: \( 1 \times 2 = 2 \), add to the next coefficient: \( -2 + 2 = 0 \). 3. Multiply by 2: \( 0 \times 2 = 0 \), add to the next coefficient: \( 7 + 0 = 7 \). 4. Multiply by 2: \( 7 \times 2 = 14 \), add to the last term: \( -14 + 14 = 0 \). The result of the division is: \[ p(x) = (x - 2)(x^2 + 7) \] ### Step 3: Factor the Quadratic (if possible) The quadratic \( x^2 + 7 \) does not factor further over the real numbers. However, if factoring over the complex numbers: \[ x^2 + 7 = (x + i\sqrt{7})(x - i\sqrt{7}) \] ### **Final Answer:** Yes, \(x - 2\) is a factor of \(p(x)\). The remaining factor is the quadratic \(x^{2} + 7\). Thus, \[ p(x) = (x - 2)(x^{2} + 7) \]