Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
