Question

if $g(v) = 2v^{4} + 16v^{3} + 25v^{2} - 33v - 20$, use synthetic division to find $g(-4)$.

Explanation:

Step1: Set up synthetic division

We are dividing by \( v = -4 \). The coefficients of the polynomial \( g(v)=2v^{4}+16v^{3}+25v^{2}-33v - 20 \) are \( 2,16,25,-33,-20 \).
We write the coefficients and the root \( - 4\) in the synthetic division setup:
\[

$$\begin{array}{r|rrrrr} -4 & 2 & 16 & 25 & -33 & -20 \\ & & & & & \\ \hline & 2 & & & & \\ \end{array}$$

\]

Step2: Bring down the leading coefficient

Bring down the first coefficient \( 2 \):
\[

$$\begin{array}{r|rrrrr} -4 & 2 & 16 & 25 & -33 & -20 \\ & & & & & \\ \hline & 2 & & & & \\ \end{array}$$

\]

Step3: Multiply and add

Multiply the number we brought down (\( 2 \)) by \( - 4 \): \( 2\times(-4)=-8 \).
Add this to the next coefficient (\( 16 \)): \( 16+(-8) = 8 \).
\[

$$\begin{array}{r|rrrrr} -4 & 2 & 16 & 25 & -33 & -20 \\ & & -8 & & & \\ \hline & 2 & 8 & & & \\ \end{array}$$

\]

Step4: Multiply and add again

Multiply the result (\( 8 \)) by \( - 4 \): \( 8\times(-4)=-32 \).
Add this to the next coefficient (\( 25 \)): \( 25+(-32)=-7 \).
\[

$$\begin{array}{r|rrrrr} -4 & 2 & 16 & 25 & -33 & -20 \\ & & -8 & -32 & & \\ \hline & 2 & 8 & -7 & & \\ \end{array}$$

\]

Step5: Multiply and add once more

Multiply the result (\( - 7 \)) by \( - 4 \): \( -7\times(-4) = 28 \).
Add this to the next coefficient (\( - 33 \)): \( -33 + 28=-5 \).
\[

$$\begin{array}{r|rrrrr} -4 & 2 & 16 & 25 & -33 & -20 \\ & & -8 & -32 & 28 & \\ \hline & 2 & 8 & -7 & -5 & \\ \end{array}$$

\]

Step6: Multiply and add the last time

Multiply the result (\( - 5 \)) by \( - 4 \): \( -5\times(-4)=20 \).
Add this to the last coefficient (\( - 20 \)): \( -20 + 20=0 \).
\[

$$\begin{array}{r|rrrrr} -4 & 2 & 16 & 25 & -33 & -20 \\ & & -8 & -32 & 28 & 20 \\ \hline & 2 & 8 & -7 & -5 & 0 \\ \end{array}$$

\]

Answer:

\( 0 \)