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Answer :
To perform the division [tex]\(x - 2 \div (x^2 + 2x + 3)\)[/tex] using synthetic division, we need to transform this into a format that we can work with since synthetic division is typically used when dividing a polynomial by a linear factor in the form [tex]\(x - c\)[/tex], which matches our divisor [tex]\(x - 2\)[/tex].
Here's how you can approach this:
1. Set up the Synthetic Division:
- Identify [tex]\(c\)[/tex] from the divisor [tex]\(x - 2\)[/tex]. In this case, [tex]\(c = 2\)[/tex].
2. Prepare the Dividend:
- Write the coefficients of the dividend [tex]\(x^2 + 2x + 3\)[/tex]. These are [tex]\(1, 2, 3\)[/tex].
3. Layout for Synthetic Division:
- Write the number [tex]\(2\)[/tex] (from your divisor [tex]\(x - 2\)[/tex]) on the left.
- List the coefficients of the dividend [tex]\(1, 2, 3\)[/tex] to the right.
```
2 | 1 2 3
|
```
4. Perform Synthetic Division Steps:
- Bring down the first coefficient (1) as is.
- Multiply this coefficient by [tex]\(2\)[/tex] and write the result under the second coefficient.
- Add this result to the second coefficient.
- Repeat this process for the rest of the coefficients.
```
2 | 1 2 3
| 2 8
--------------
1 4 11
```
5. Interpret the Result:
- The numbers on the bottom (after addition) represent the coefficients of the quotient polynomial, plus a remainder.
- So, we have a quotient [tex]\(1x + 4\)[/tex] and a remainder of [tex]\(11\)[/tex].
Therefore, the quotient is [tex]\(x + 4\)[/tex] and the remainder is 11. We can write the division as:
[tex]\[
x^2 + 2x + 3 = (x - 2)(x + 4) + 11
\][/tex]
The blank that you need to fill with a number corresponding to a step in the process likely relates to one of these key numbers from the synthetic division, such as the remainder or one of the coefficients in the intermediate steps. The given answer "23" does not appear to directly correlate to a needed step in the synthetic division shown, possibly a misunderstanding or calculation representation. In traditional terms, a common result linked to synthetic division would be numbers like coefficients or remainders closely tied to the polynomial operation.
Here's how you can approach this:
1. Set up the Synthetic Division:
- Identify [tex]\(c\)[/tex] from the divisor [tex]\(x - 2\)[/tex]. In this case, [tex]\(c = 2\)[/tex].
2. Prepare the Dividend:
- Write the coefficients of the dividend [tex]\(x^2 + 2x + 3\)[/tex]. These are [tex]\(1, 2, 3\)[/tex].
3. Layout for Synthetic Division:
- Write the number [tex]\(2\)[/tex] (from your divisor [tex]\(x - 2\)[/tex]) on the left.
- List the coefficients of the dividend [tex]\(1, 2, 3\)[/tex] to the right.
```
2 | 1 2 3
|
```
4. Perform Synthetic Division Steps:
- Bring down the first coefficient (1) as is.
- Multiply this coefficient by [tex]\(2\)[/tex] and write the result under the second coefficient.
- Add this result to the second coefficient.
- Repeat this process for the rest of the coefficients.
```
2 | 1 2 3
| 2 8
--------------
1 4 11
```
5. Interpret the Result:
- The numbers on the bottom (after addition) represent the coefficients of the quotient polynomial, plus a remainder.
- So, we have a quotient [tex]\(1x + 4\)[/tex] and a remainder of [tex]\(11\)[/tex].
Therefore, the quotient is [tex]\(x + 4\)[/tex] and the remainder is 11. We can write the division as:
[tex]\[
x^2 + 2x + 3 = (x - 2)(x + 4) + 11
\][/tex]
The blank that you need to fill with a number corresponding to a step in the process likely relates to one of these key numbers from the synthetic division, such as the remainder or one of the coefficients in the intermediate steps. The given answer "23" does not appear to directly correlate to a needed step in the synthetic division shown, possibly a misunderstanding or calculation representation. In traditional terms, a common result linked to synthetic division would be numbers like coefficients or remainders closely tied to the polynomial operation.
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