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Fill in the blanks to correctly complete the sentence.

To perform the division [tex]x - 1 \longdiv { x ^ { 3 } + 2 x ^ { 2 } + 4 x }[/tex], begin by writing the synthetic division problem shown below:

[tex]
\[
\begin{array}{r|rrrr}
1 & \square & 2 & \square & 4 & \square \\
\end{array}
\]
[/tex]

Answer :

Of course! Let's go through the process of synthetic division to divide [tex]\( x^3 + 2x^2 + 4x \)[/tex] by [tex]\( x - 1 \)[/tex].

### Step-by-Step Solution

1. Set up the synthetic division:

- The divisor is [tex]\( x - 1 \)[/tex], so we use 1 as our synthetic divisor.
- The coefficients of the dividend [tex]\( x^3 + 2x^2 + 4x \)[/tex] are [tex]\( 1, 2, 0, 4 \)[/tex].

2. Write the coefficients of the dividend:

```
1 2 4 0
```

3. Perform the synthetic division process:

- Write the synthetic divisor (1) to the left:

```
1 | 1 2 4 0
```

- Bring down the leading coefficient (first number in the row) directly below the line:

```
1 | 1 2 4 0
|_____________
1
```

- Multiply the divisor (1) by this result (1), and write the product under the next coefficient:

```
1 | 1 2 4 0
| 1
|_____________
1
```

- Add the numbers in the second column: [tex]\( 2 + 1 = 3 \)[/tex].

```
1 | 1 2 4 0
| 1
|_____________
1 3
```

- Repeat the process: multiply 1 by the new result (3) and write it under the next coefficient:

```
1 | 1 2 4 0
| 1 3
|_____________
1 3 4
```

- Add: [tex]\( 4 + 3 = 7 \)[/tex].

```
1 | 1 2 4 0
| 1 3 7
|_____________
1 3 7 7
```

4. Conclusion:

- The bottom row (excluding the last number) represents the coefficients of the quotient:
- Quotient: [tex]\( 1x^2 + 3x + 7 \)[/tex]
- The last number is the remainder:
- Remainder: [tex]\( 7 \)[/tex]

So, the synthetic division results in:
[tex]\[ x^2 + 3x + 7 + \frac{7}{x-1} \][/tex]

### Final Synthetic Division Setup:
```
1 | 1 2 4 0
| 1 3 7
|________________
1 3 7 (7)
```

The filled-in sentence should be:
"To perform the division [tex]\( x - 1 \longdiv { x ^ { 3 } + 2 x ^ { 2 } + 4 x }\)[/tex], begin by writing the synthetic division problem shown below 1 1 3 7."

I hope this helps! Let me know if you have any other questions.

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