Answer :
Sure! Let's tackle the problem step-by-step to understand how to perform the division [tex]\( x + 1 \div (x^3 + 4x + 2) \)[/tex].
We begin by performing synthetic division. Here's a detailed breakdown:
1. Identify the dividend and the divisor:
- Dividend: [tex]\( x^3 + 4x + 2 \)[/tex]
- Divisor: [tex]\( x + 1 \)[/tex]
2. Write the coefficients of the dividend in order of descending powers of [tex]\( x \)[/tex]. Make sure to include zeros for missing terms:
- The polynomial [tex]\( x^3 + 0x^2 + 4x + 2 \)[/tex] has coefficients: [tex]\([1, 0, 4, 2]\)[/tex].
3. For synthetic division, we need to use the root of the divisor polynomial. Since [tex]\( x + 1 = 0 \)[/tex] gives [tex]\( x = -1 \)[/tex], we will use [tex]\(-1\)[/tex] in our synthetic division process.
4. Set up the synthetic division as follows:
- Write down the coefficients: [tex]\( 1, 0, 4, 2 \)[/tex].
- Write the root of the divisor (which is [tex]\(-1\)[/tex]) to the left.
Here's the setup step-by-step:
```
-1 | 1 0 4 2
|_________________
```
5. Now, perform the synthetic division process:
- Start by bringing down the first coefficient (1) straight down:
```
-1 | 1 0 4 2
| (1)
|_________________
1
```
- Multiply [tex]\(-1\)[/tex] (the root) by the number below the line (1) and write the result under the next coefficient:
```
-1 | 1 0 4 2
| -1
|_________________
1 -1
```
- Add the numbers in the column: [tex]\( 0 + (-1) = -1 \)[/tex]
```
-1 | 1 0 4 2
| -1 5
|_________________
1 -1 5
```
- Repeat the process: multiply [tex]\(-1\)[/tex] by [tex]\(-1\)[/tex] and add it to [tex]\(4\)[/tex]:
```
-1 | 1 0 4 2
| -1 5
|_________________
1 -1 3
```
- Finally, multiply [tex]\(-1\)[/tex] by [tex]\(3\)[/tex] and add it to [tex]\(2\)[/tex]:
```
-1 | 1 0 4 2
| -1 5
|_________________
1 -1 3 -1
```
Now, we have completed the synthetic division.
The result of the synthetic division provides the coefficients of the quotient and the remainder. The bottom row (excluding the divisor's root) represents these coefficients. Therefore, the quotient is [tex]\( x^2 - x + 3 \)[/tex] and the remainder is [tex]\(-1\)[/tex].
In summary, the synthetic division setup began with the following:
[tex]\[ 1 \][/tex]
So, to fill in the blanks correctly to complete the sentence:
"To perform the division [tex]\( x + 1 \div (x^3 + 4x + 2) \)[/tex], begin by writing the synthetic division problem shown below:
1 [tex]\(\square\)[/tex]
42"
The correct step is:
1
Thus, the blank can be filled in as follows:
1 (in the blank space above the 42).
We begin by performing synthetic division. Here's a detailed breakdown:
1. Identify the dividend and the divisor:
- Dividend: [tex]\( x^3 + 4x + 2 \)[/tex]
- Divisor: [tex]\( x + 1 \)[/tex]
2. Write the coefficients of the dividend in order of descending powers of [tex]\( x \)[/tex]. Make sure to include zeros for missing terms:
- The polynomial [tex]\( x^3 + 0x^2 + 4x + 2 \)[/tex] has coefficients: [tex]\([1, 0, 4, 2]\)[/tex].
3. For synthetic division, we need to use the root of the divisor polynomial. Since [tex]\( x + 1 = 0 \)[/tex] gives [tex]\( x = -1 \)[/tex], we will use [tex]\(-1\)[/tex] in our synthetic division process.
4. Set up the synthetic division as follows:
- Write down the coefficients: [tex]\( 1, 0, 4, 2 \)[/tex].
- Write the root of the divisor (which is [tex]\(-1\)[/tex]) to the left.
Here's the setup step-by-step:
```
-1 | 1 0 4 2
|_________________
```
5. Now, perform the synthetic division process:
- Start by bringing down the first coefficient (1) straight down:
```
-1 | 1 0 4 2
| (1)
|_________________
1
```
- Multiply [tex]\(-1\)[/tex] (the root) by the number below the line (1) and write the result under the next coefficient:
```
-1 | 1 0 4 2
| -1
|_________________
1 -1
```
- Add the numbers in the column: [tex]\( 0 + (-1) = -1 \)[/tex]
```
-1 | 1 0 4 2
| -1 5
|_________________
1 -1 5
```
- Repeat the process: multiply [tex]\(-1\)[/tex] by [tex]\(-1\)[/tex] and add it to [tex]\(4\)[/tex]:
```
-1 | 1 0 4 2
| -1 5
|_________________
1 -1 3
```
- Finally, multiply [tex]\(-1\)[/tex] by [tex]\(3\)[/tex] and add it to [tex]\(2\)[/tex]:
```
-1 | 1 0 4 2
| -1 5
|_________________
1 -1 3 -1
```
Now, we have completed the synthetic division.
The result of the synthetic division provides the coefficients of the quotient and the remainder. The bottom row (excluding the divisor's root) represents these coefficients. Therefore, the quotient is [tex]\( x^2 - x + 3 \)[/tex] and the remainder is [tex]\(-1\)[/tex].
In summary, the synthetic division setup began with the following:
[tex]\[ 1 \][/tex]
So, to fill in the blanks correctly to complete the sentence:
"To perform the division [tex]\( x + 1 \div (x^3 + 4x + 2) \)[/tex], begin by writing the synthetic division problem shown below:
1 [tex]\(\square\)[/tex]
42"
The correct step is:
1
Thus, the blank can be filled in as follows:
1 (in the blank space above the 42).
