Use synthetic division to find the quotient and remainder when [tex]-x^3 - 7x^2 + 6[/tex] is divided by [tex]x + 7[/tex] by completing the parts below.

(a) Complete this synthetic division table.

(b) Write your answer in the following form: Quotient [tex]+ \frac{\text{Remainder}}{x+7}[/tex].

[tex]
\frac{-x^3 - 7x^2 + 6}{x + 7} = \square + \frac{\square}{x + 7}
[/tex]



Answer :

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]