91Ó°ÊÓ

To find \(f(c)\) directly, substitute \(c = 3\) into the function \(f(n) = 3n^4 - 2n^3 + 4n - 1\). Compute as follows:\[f(3) = 3(3)^4 - 2(3)^3 + 4(3) - 1\]First, calculate \(3^4 = 81\) and \(3^3 = 27\). Then, substitute back in:\[f(3) = 3 \times 81 - 2 \times 27 + 4 \times 3 - 1\]Calculate each term:\[f(3) = 243 - 54 + 12 - 1\]Adding these together gives:\[f(3) = 200\]

Write the coefficients of the polynomial \(3n^4 - 2n^3 + 0n^2 + 4n - 1\) in order: \([3, -2, 0, 4, -1]\). Use \(c = 3\) for synthetic division. Begin by bringing down the first coefficient (3):\[\begin{array}{r|rrrrr}3 & 3 & -2 & 0 & 4 & -1 \ & & & & & \\hline & 3 & & & & \\end{array}\]

Continue synthetic division by multiplying 3 by the current bottom number and adding it to the next coefficient:1. Multiply 3 (from below) by 3 (coefficient) and add to -2: \[ \begin{array}{r|rrrrr} 3 & 3 & -2 & 0 & 4 & -1 \ & & 9 & & & \\hline & 3 & 7 & & & \ \end{array} \]2. Multiply 3 by 7 and add to 0: \[ \begin{array}{r|rrrrr} 3 & 3 & -2 & 0 & 4 & -1 \ & & 9 & 21 & & \\hline & 3 & 7 & 21 & & \ \end{array} \]3. Multiply 3 by 21 and add to 4: \[ \begin{array}{r|rrrrr} 3 & 3 & -2 & 0 & 4 & -1 \ & & 9 & 21 & 63 & \\hline & 3 & 7 & 21 & 67 & \ \end{array} \]4. Multiply 3 by 67 and add to -1: \[ \begin{array}{r|rrrrr} 3 & 3 & -2 & 0 & 4 & -1 \ & & 9 & 21 & 63 &201\\hline & 3 & 7 & 21 & 67 &200\ \end{array} \]

The remainder from the synthetic division is 200, which matches the direct evaluation \(f(3) = 200\). By the Remainder Theorem, the remainder obtained from synthetic division when dividing \(f(n)\) by \(n-c\) is \(f(c)\). Thus, this verifies that \(f(3) = 200\).