To perform synthetic division for the polynomial [tex]$3x^3 - 25x^2 + 12x - 32$[/tex] by the divisor [tex]$x - 8$[/tex], follow these steps:
1. Identify the root of the divisor [tex]$x - 8$[/tex], which is [tex]$8$[/tex].
2. Write down the coefficients of the polynomial: [tex]$3, -25, 12, -32$[/tex].
3. Start with the leading coefficient, [tex]$3$[/tex], and bring it down.
Now, proceed with the synthetic division process step by step:
[tex]\[
\begin{array}{r|rrrr}
8 & 3 & -25 & 12 & -32 \\
& & & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
4. Multiply the value just written down by the root [tex]$8$[/tex] and write the result under the next coefficient.
[tex]\[
\begin{array}{r|rrrr}
8 & 3 & -25 & 12 & -32 \\
& & 24 & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
5. Add the column: [tex]$-25 + 24 = -1$[/tex]. Write this result in the bottom row.
[tex]\[
\begin{array}{r|rrrr}
8 & 3 & -25 & 12 & -32 \\
& & 24 & & \\
\hline
& 3 & -1 & & \\
\end{array}
\][/tex]
6. Repeat the process: Multiply [tex]$-1$[/tex] by [tex]$8$[/tex] (write [tex]$-8$[/tex] under the next coefficient), then add the column.
[tex]\[
\begin{array}{r|rrrr}
8 & 3 & -25 & 12 & -32 \\
& & 24 & -8 & \\
\hline
& 3 & -1 & 4 & \\
\end{array}
\][/tex]
7. Continue: Multiply [tex]$4$[/tex] by [tex]$8$[/tex] (write [tex]$32$[/tex] under the last coefficient), then add the column.
[tex]\[
\begin{array}{r|rrrr}
8 & 3 & -25 & 12 & -32 \\
& & 24 & -8 & 32 \\
\hline
& 3 & -1 & 4 & 0 \\
\end{array}
\][/tex]
8. The final row gives the coefficients of the quotient polynomial and the remainder. The quotient is [tex]$3x^2 - x + 4$[/tex], and the remainder is [tex]$0$[/tex].
Thus, the quotient of the division of [tex]$3x^3 - 25x^2 + 12x - 32$[/tex] by [tex]$x - 8$[/tex] is [tex]$3x^2 - x + 4$[/tex], and the remainder is [tex]$0$[/tex].
