Let's perform synthetic division to find the quotient and remainder when [tex]\(-x^3 - 7x^2 + 6\)[/tex] is divided by [tex]\(x + 7\)[/tex].
### Step-by-Step Solution:
#### Part (a): Complete the Synthetic Division Table
1. Identify the root for synthetic division. Since we are dividing by [tex]\(x + 7\)[/tex], the root is [tex]\(-7\)[/tex].
2. Write down the coefficients of the polynomial [tex]\(-x^3 - 7x^2 + 6\)[/tex]. The coefficients are: [tex]\(-1, -7, 0, 6\)[/tex] (noting the [tex]\(0\)[/tex] coefficient for the missing [tex]\(x\)[/tex] term).
3. Set up the synthetic division table with these coefficients.
To visualize the synthetic division:
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & 7 & 0 & 0 \\
\hline
& -1 & 0 & 0 & 6 \\
\end{array}
\][/tex]
#### Steps:
1. Bring down the first coefficient [tex]\(-1\)[/tex].
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & & & \\
\hline
& -1 & & & \\
\end{array}
\][/tex]
2. Multiply [tex]\(-7\)[/tex] (root) by [tex]\(-1\)[/tex] (value just written) and place the result under the next coefficient.
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & 7 & & \\
\hline
& -1 & 0 & & \\
\end{array}
\][/tex]
3. Add the numbers in the second column: [tex]\(-7 + 7 = 0\)[/tex].
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & 7 & 0 & \\
\hline
& -1 & 0 & 0 & \\
\end{array}
\][/tex]
4. Multiply [tex]\(-7\)[/tex] by [tex]\(0\)[/tex] (value just written) and place the result under the next coefficient.
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & 7 & 0 & 0 \\
\hline
& -1 & 0 & 0 & \\
\end{array}
\][/tex]
5. Add the numbers in the third column: [tex]\(0 + 0 = 0\)[/tex].
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & 7 & 0 & 0 \\
\hline
& -1 & 0 & 0 & 6 \\
\end{array}
\][/tex]
6. Multiply [tex]\(-7\)[/tex] by [tex]\(0\)[/tex] (value just written) and place the result under the next coefficient.
[tex]\[
\begin{array}{r|rrrr}
-7 & -1 & -7 & 0 & 6 \\
& & 7 & 0 & 0 \\
\hline
& -1 & 0 & 0 & 6 \\
\end{array}
\][/tex]
7. Add the numbers in the fourth column: [tex]\(6 + 0 = 6\)[/tex].
The division process is complete, and the final row gives the quotient coefficients and the remainder.
#### Part (b): Write Your Answer in the Required Form
The quotient is formed from the first three numbers in the bottom row, which correspond to the coefficients of the quotient polynomial. The remainder is the last number in the bottom row.
[tex]\[
\text{Quotient: } -1x^2 + 0x + 0 = -x^2
\][/tex]
[tex]\[
\text{Remainder: } 6
\][/tex]
Combining these results, the division of [tex]\(-x^3 - 7x^2 + 6\)[/tex] by [tex]\(x + 7\)[/tex] yields:
[tex]\[
\frac{-x^3 - 7x^2 + 6}{x + 7} = -x^2 + \frac{6}{x + 7}
\][/tex]
So, the final answer in the required form is:
[tex]\[
\frac{-x^3 - 7x^2 + 6}{x + 7} = -x^2 + \frac{6}{x + 7}
\][/tex]
