+26367 What is the value of
\(k\) if \(x^3 - kx^2 + 2x + 1\) is divided by \(x - 4\)
and gives a remainder of \( -7\) ?
polynomial long division:
\(x^3 - kx^2 + 2x + 1 : x - 4 = x^2 - x·k + 4·x - 4·k + 18 - \dfrac{16·k - 73}{x - 4}\)
remainder is -7:
\(\begin{array}{|rcll|} \hline -(16·k - 73) &=& -7 \\ 16·k - 73 &=& 7 \\ 16·k &=& 7+73 \\ 16·k &=& 80 \\ \mathbf{ k } & \mathbf{=} & \mathbf{5} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline x^3 - 5x^2 + 2x + 1 : x-4 &=& x^2 - x - 2 -7 \\ \hline \end{array}\)
or:
\(\begin{array}{|rcll|} \hline P(4) = -7 \\\\ P(4) = 4^3-k\cdot 4^2 + 2\cdot 4 + 1 &=& - 7 \\ 64-16k + 8 + 1 &=& - 7 \\ 73-16k &=& - 7 \\ 16k &=&73+7 \\ 16k &=& 80 \\ \mathbf{ k } & \mathbf{=} & \mathbf{5} \\ \hline \end{array}\)
