Absolutely! Let's dive into this problem step-by-step using synthetic division.
Given the polynomial [tex]\(-x^3 - 7x^2 + 6\)[/tex] and we want to divide it by [tex]\(x + 7\)[/tex].
### Step 1: Set up the synthetic division table
For synthetic division, we use the root of the divisor [tex]\(x + 7 = 0\)[/tex], which gives [tex]\(x = -7\)[/tex].
The coefficients of the polynomial [tex]\(-x^3 - 7x^2 + 0x + 6\)[/tex] are:
[tex]\[ a_3 = -1, \; a_2 = -7, \; a_1 = 0, \; a_0 = 6 \][/tex]
We are to divide these coefficients using [tex]\(x = -7\)[/tex].
### Step 2: Perform synthetic division
Set up the table:
[tex]\[
\begin{array}{c|cccc}
-7 & -1 & -7 & 0 & 6 \\
\hline
& & & & \\
& -1 & & & \\
\end{array}
\][/tex]
First column:
- Bring down the first coefficient as it is:
[tex]\[
-1
\][/tex]
Second column:
- Multiply the first coefficient by [tex]\(-7\)[/tex] and add to the second coefficient:
[tex]\[
-7 \times -1 = 7 \quad \Rightarrow \quad -7 + 7 = 0
\][/tex]
[tex]\[
\begin{array}{c|cccc}
-7 & -1 & -7 & 0 & 6 \\
\hline
& & 0 & & \\
-1 & 0 & & \\
\end{array}
\][/tex]
Third column:
- Multiply the result from the second step by [tex]\(-7\)[/tex] and add to the third coefficient:
[tex]\[
-7 \times 0 = 0 \quad \Rightarrow \quad 0 + 0 = 0
\][/tex]
[tex]\[
\begin{array}{c|cccc}
-7 & -1 & -7 & 0 & 6 \\
\hline
& & & 0 & \\
-1 & 0 & 0 & \\
\end{array}
\][/tex]
Fourth column:
- Multiply the result from the third step by [tex]\(-7\)[/tex] and add to the fourth coefficient:
[tex]\[
-7 \times 0 = 0 \quad \Rightarrow \quad 6 + 0 = 6
\][/tex]
[tex]\[
\begin{array}{c|cccc}
-7 & -1 & -7 & 0 & 6 \\
\hline
& & & & 6 \\
-1 & 0 & 0 & 6 \\
\end{array}
\][/tex]
### Step 3: Interpret the results
Quotient:
The coefficients from synthetic division for the quotient polynomial are:
[tex]\[
-1, \; 0, \; 0 \quad \Rightarrow \quad -x^2 + 0x + 0 \quad \Rightarrow \quad -x^2
\][/tex]
Remainder:
The last value is the remainder:
[tex]\[
6
\][/tex]
### Final Answer:
Combining the quotient and the remainder, we have:
[tex]\[
\frac{-x^3 - 7x^2 + 6}{x + 7} = -x^2 + \frac{6}{x + 7}
\][/tex]
Therefore, the solution in the form asked is:
[tex]\[
\boxed{-x^2 + \frac{6}{x+7}}
\][/tex]
