Question

if $p(x) = 3x^3 + 7x^2 - 5x - 25$, use synthetic division to find $p(-3)$.

Explanation:

Step1: Set up synthetic division

For \( p(x) = 3x^3 + 7x^2 - 5x - 25 \) and \( x=-3 \), the coefficients are \( 3, 7, -5, -25 \). Write them in a row and put \( -3 \) outside the division bracket.
\[

$$\begin{array}{r|rrrr} -3 & 3 & 7 & -5 & -25 \\ & & & & \\ \hline & & & & \\ \end{array}$$

\]

Step2: Bring down the leading coefficient

Bring down the \( 3 \) to the bottom row.
\[

$$\begin{array}{r|rrrr} -3 & 3 & 7 & -5 & -25 \\ & & & & \\ \hline & 3 & & & \\ \end{array}$$

\]

Step3: Multiply and add

Multiply the brought - down number (\( 3 \)) by \( -3 \): \( 3\times(-3)=-9 \). Add this to the next coefficient (\( 7 \)): \( 7 + (-9)=-2 \).
\[

$$\begin{array}{r|rrrr} -3 & 3 & 7 & -5 & -25 \\ & & -9 & & \\ \hline & 3 & -2 & & \\ \end{array}$$

\]

Step4: Multiply and add again

Multiply the new number (\( -2 \)) by \( -3 \): \( -2\times(-3) = 6 \). Add this to the next coefficient (\( -5 \)): \( -5+6 = 1 \).
\[

$$\begin{array}{r|rrrr} -3 & 3 & 7 & -5 & -25 \\ & & -9 & 6 & \\ \hline & 3 & -2 & 1 & \\ \end{array}$$

\]

Step5: Multiply and add the last time

Multiply the new number (\( 1 \)) by \( -3 \): \( 1\times(-3)=-3 \). Add this to the last coefficient (\( -25 \)): \( -25+(-3)=-28 \).
\[

$$\begin{array}{r|rrrr} -3 & 3 & 7 & -5 & -25 \\ & & -9 & 6 & -3 \\ \hline & 3 & -2 & 1 & -28 \\ \end{array}$$

\]
The last number in the bottom row is the value of \( p(-3) \).

Answer:

\( -28 \)