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:

$$
LATEXBLOCK0
$$

Step2: Bring down the leading coefficient

Bring down the first coefficient \( 2 \):

$$
LATEXBLOCK1
$$

Step3: Multiply and add

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

$$
LATEXBLOCK2
$$

Step4: Multiply and add again

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

$$
LATEXBLOCK3
$$

Step5: Multiply and add once more

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

$$
LATEXBLOCK4
$$

Step6: Multiply and add the last time

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

$$
LATEXBLOCK5
$$

Answer:

\( 0 \)