Given the function \( f(x)=7 x^{2}-4 x+2 \). Calculate the following values using synthetic division and the Remainder Theorem: \( f(-2)=\square \) \( f(-1)=\square \) \( f(0)=\square \) \( f(1)=\square \) \( f(2)=\square \)

To calculate the values of the function \( f(x) = 7x^2 - 4x + 2 \) at the specified \( x \) values using synthetic division and the Remainder Theorem, we perform the following steps: 1. **\( f(-2) \):** - Coefficients for synthetic division are \( [7, -4, 2] \). - Using \(-2\): \[ \begin{array}{r|rrr} -2 & 7 & -4 & 2 \\ & & -14 & 36 \\ \hline & 7 & -18 & 38 \\ \end{array} \] The remainder is \(38\), so \( f(-2) = 38 \). 2. **\( f(-1) \):** - Using \(-1\): \[ \begin{array}{r|rrr} -1 & 7 & -4 & 2 \\ & & -7 & 11 \\ \hline & 7 & -11 & 13 \\ \end{array} \] The remainder is \(13\), so \( f(-1) = 13 \). 3. **\( f(0) \):** - For \( f(0) \), we simply substitute \( x = 0 \): \[ f(0) = 7(0)^2 - 4(0) + 2 = 2. \] 4. **\( f(1) \):** - Using \(1\): \[ \begin{array}{r|rrr} 1 & 7 & -4 & 2 \\ & & 7 & 3 \\ \hline & 7 & 3 & 5 \\ \end{array} \] The remainder is \(5\), so \( f(1) = 5 \). 5. **\( f(2) \):** - Using \(2\): \[ \begin{array}{r|rrr} 2 & 7 & -4 & 2 \\ & & 14 & 20 \\ \hline & 7 & 10 & 22 \\ \end{array} \] The remainder is \(22\), so \( f(2) = 22 \). Putting these results together we get: - \( f(-2) = 38 \) - \( f(-1) = 13 \) - \( f(0) = 2 \) - \( f(1) = 5 \) - \( f(2) = 22 \) Thus, the final results are: - \( f(-2) = 38 \) - \( f(-1) = 13 \) - \( f(0) = 2 \) - \( f(1) = 5 \) - \( f(2) = 22 \)