Answer :
To determine which of the given options shows the division problem in synthetic division form, we need to understand how synthetic division works and apply it properly to the given polynomial [tex]\(\frac{7x^2 - 2x + 4}{x + 3}\)[/tex].
### Step-by-Step Solution:
1. Identify the Divisor:
- The divisor is [tex]\( x + 3 \)[/tex]. For synthetic division, we use the zero of the divisor, which is [tex]\( x = -3 \)[/tex].
2. Set up the Synthetic Division:
- Write down the coefficients of the polynomial [tex]\(7x^2 - 2x + 4\)[/tex]. These coefficients are: [tex]\( 7, -2, 4 \)[/tex].
3. Perform Synthetic Division:
- Start by writing the zero of the divisor (which is [tex]\(-3\)[/tex]) to the left, and the coefficients of the polynomial to the right.
```
-3 | 7 -2 4
```
4. Carry Down the First Coefficient:
- Bring down the first coefficient (7) as it is.
```
-3 | 7 -2 4
|
----------------
7
```
5. Multiply and Add:
- Multiply [tex]\(-3\)[/tex] by 7 and place the result under the next coefficient.
- Add this result to the next coefficient ([tex]\(-2\)[/tex]) and place the sum below the line.
```
-3 | 7 -2 4
| -21
----------------
7 -23
```
- Repeat the process:
- Multiply [tex]\(-3\)[/tex] by [tex]\(-23\)[/tex] and place the result under the next coefficient.
- Add this result to the next coefficient (4) and place the sum below the line.
```
-3 | 7 -2 4
| -21 69
----------------
7 -23 73
```
6. The Result:
- The numbers below the line ([tex]\(7, -23, 73\)[/tex]) represent the coefficients of the quotient polynomial and the remainder.
Now, recognize the correct synthetic division setup from the choices given.
Comparing the options:
- Option B: [tex]\( 3 \longdiv { 7 \quad 2 \quad 4 }\)[/tex]
- This uses the value [tex]\(3\)[/tex] as the divisor zero and has a wrong coefficient for the middle term (should be [tex]\(-2\)[/tex] instead of [tex]\(2\)[/tex]).
- Option D: [tex]\(3 \left\lvert\, \begin{array}{lll}7 & -2 & 4\end{array}\right.\)[/tex]
- This correctly sets up the coefficients and divisor zero for synthetic division.
Therefore, the correct option that shows the division problem [tex]\( \frac{7x^2 - 2x + 4}{x + 3} \)[/tex] in synthetic division form is:
[tex]\[ \boxed{D} \][/tex]
### Step-by-Step Solution:
1. Identify the Divisor:
- The divisor is [tex]\( x + 3 \)[/tex]. For synthetic division, we use the zero of the divisor, which is [tex]\( x = -3 \)[/tex].
2. Set up the Synthetic Division:
- Write down the coefficients of the polynomial [tex]\(7x^2 - 2x + 4\)[/tex]. These coefficients are: [tex]\( 7, -2, 4 \)[/tex].
3. Perform Synthetic Division:
- Start by writing the zero of the divisor (which is [tex]\(-3\)[/tex]) to the left, and the coefficients of the polynomial to the right.
```
-3 | 7 -2 4
```
4. Carry Down the First Coefficient:
- Bring down the first coefficient (7) as it is.
```
-3 | 7 -2 4
|
----------------
7
```
5. Multiply and Add:
- Multiply [tex]\(-3\)[/tex] by 7 and place the result under the next coefficient.
- Add this result to the next coefficient ([tex]\(-2\)[/tex]) and place the sum below the line.
```
-3 | 7 -2 4
| -21
----------------
7 -23
```
- Repeat the process:
- Multiply [tex]\(-3\)[/tex] by [tex]\(-23\)[/tex] and place the result under the next coefficient.
- Add this result to the next coefficient (4) and place the sum below the line.
```
-3 | 7 -2 4
| -21 69
----------------
7 -23 73
```
6. The Result:
- The numbers below the line ([tex]\(7, -23, 73\)[/tex]) represent the coefficients of the quotient polynomial and the remainder.
Now, recognize the correct synthetic division setup from the choices given.
Comparing the options:
- Option B: [tex]\( 3 \longdiv { 7 \quad 2 \quad 4 }\)[/tex]
- This uses the value [tex]\(3\)[/tex] as the divisor zero and has a wrong coefficient for the middle term (should be [tex]\(-2\)[/tex] instead of [tex]\(2\)[/tex]).
- Option D: [tex]\(3 \left\lvert\, \begin{array}{lll}7 & -2 & 4\end{array}\right.\)[/tex]
- This correctly sets up the coefficients and divisor zero for synthetic division.
Therefore, the correct option that shows the division problem [tex]\( \frac{7x^2 - 2x + 4}{x + 3} \)[/tex] in synthetic division form is:
[tex]\[ \boxed{D} \][/tex]
