Questions: Use the remainder theorem and synthetic division to find f(k) for the given value of k. f(x)=-x^3-8x^2-17x-6 ; k=-4 f(-4)=

Transcript text: Use the remainder theorem and synthetic division to find $f(k)$ for the given value of $k$. \[ \begin{array}{l} f(x)=-x^{3}-8 x^{2}-17 x-6 ; k=-4 \\ f(-4)=\square \end{array} \]

Solution

Solution Steps

Step 1: Set up the Synthetic Division Table

To evaluate $f(k)$ for a polynomial $f(x)$ using synthetic division, we first set up a table. The coefficients of $f(x)$ are listed, and the value of $k$ is used to perform the division. The process involves multiplying and adding through the coefficients with the value of $k$ to iteratively compute the next value in the table.

Step 2: Perform the Synthetic Division

Starting with the first coefficient, each subsequent coefficient is processed by multiplying the last result by $k$ and adding the current coefficient. This process is repeated until all coefficients have been processed. The final value obtained is the remainder, which, according to the Remainder Theorem, equals $f(k)$. The coefficients used are [-1, -8, -17, -6], and the value of $k$ is -4.