Let's go through the synthetic division step-by-step to find the quotient and remainder when dividing [tex]\(x^3 - 10x^2 + 12x + 3\)[/tex] by [tex]\(x - 2\)[/tex].
### (a) Complete the Synthetic Division Table
#### Step 1: Set up the synthetic division table.
Place the root of the divisor (i.e., for [tex]\(x - 2\)[/tex], the root is 2) to the left and the coefficients of the polynomial to the right:
[tex]\[
\begin{array}{r|cccc}
2 & 1 & -10 & 12 & 3 \\
\end{array}
\][/tex]
#### Step 2: Bring down the leading coefficient.
[tex]\[
\begin{array}{r|cccc}
2 & 1 & -10 & 12 & 3 \\
\hline
& 1 & & & \\
\end{array}
\][/tex]
#### Step 3: Multiply the root by the value just brought down and add to the next coefficient.
1. Multiply [tex]\(2 \times 1 = 2\)[/tex]. Add this to the next coefficient [tex]\(-10\)[/tex]:
[tex]\[
-10 + 2 = -8
\][/tex]
[tex]\[
\begin{array}{r|cccc}
2 & 1 & -10 & 12 & 3 \\
\hline
& 1 & -8 & & \\
\end{array}
\][/tex]
2. Multiply [tex]\(2 \times -8 = -16\)[/tex]. Add this to the next coefficient [tex]\(12\)[/tex]:
[tex]\[
12 + (-16) = -4
\][/tex]
[tex]\[
\begin{array}{r|cccc}
2 & 1 & -10 & 12 & 3 \\
\hline
& 1 & -8 & -4 & \\
\end{array}
\][/tex]
3. Multiply [tex]\(2 \times -4 = -8\)[/tex]. Add this to the next coefficient [tex]\(3\)[/tex]:
[tex]\[
3 + (-8) = -5
\][/tex]
[tex]\[
\begin{array}{r|cccc}
2 & 1 & -10 & 12 & 3 \\
\hline
& 1 & -8 & -4 & -5 \\
\end{array}
\][/tex]
Thus, the completed synthetic division table is:
[tex]\[
\begin{array}{r|cccc}
2 & 1 & -10 & 12 & 3 \\
\hline
& 1 & -8 & -4 & -5 \\
\end{array}
\][/tex]
The coefficients of the quotient are [tex]\(1, -8, -4\)[/tex], and the remainder is [tex]\(-5\)[/tex].
### (b) Write the result in the form [tex]\( \text{Quotient} + \frac{\text{Remainder}}{x-2} \)[/tex].
The quotient is formed by the coefficients [tex]\(1, -8, -4\)[/tex]:
[tex]\[
x^2 - 8x - 4
\][/tex]
The remainder is [tex]\(-5\)[/tex]. Therefore, the division can be written as:
[tex]\[
\frac{x^3 - 10x^2 + 12x + 3}{x - 2} = x^2 - 8x - 4 + \frac{-5}{x-2}
\][/tex]
So, we have:
[tex]\[
\frac{x^3 - 10x^2 + 12x + 3}{x - 2} = x^2 - 8x - 4 + \frac{-5}{x-2}
\][/tex]
