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Answer :
To solve this synthetic division problem, we'll divide the polynomial by using synthetic division. Here are the steps:
1. Identify the divisor: We have the divisor [tex]\( x - 2 \)[/tex], which means we use the root [tex]\( 2 \)[/tex] in synthetic division.
2. Set up synthetic division: Write down the coefficients of the polynomial. In this case, the polynomial is represented by [tex]\( 1x + 5 \)[/tex]. So, the coefficients are [tex]\( 1 \)[/tex] and [tex]\( 5 \)[/tex].
3. Perform synthetic division:
- Step 1: Bring down the leading coefficient [tex]\( 1 \)[/tex].
- Step 2: Multiply this by the root [tex]\( 2 \)[/tex] and write the result under the next coefficient [tex]\( 5 \)[/tex].
- Step 3: Add the numbers in the second column: [tex]\( 5 + (2 \times 1) = 7 \)[/tex].
The setup looks like this:
```
2 | 1 5
| 2
------------
1 7
```
4. Interpret the result: After carrying out the division, the numbers at the bottom represent the coefficients of the quotient polynomial.
- The [tex]\( 1 \)[/tex] is the coefficient of [tex]\( x \)[/tex].
- The [tex]\( 7 \)[/tex] is the constant term.
5. Write the quotient in polynomial form: The result is [tex]\( x + 7 \)[/tex].
Therefore, the quotient is [tex]\( x + 7 \)[/tex].
Thus, the correct answer is [tex]\( \text{D. } x + 7 \)[/tex].
1. Identify the divisor: We have the divisor [tex]\( x - 2 \)[/tex], which means we use the root [tex]\( 2 \)[/tex] in synthetic division.
2. Set up synthetic division: Write down the coefficients of the polynomial. In this case, the polynomial is represented by [tex]\( 1x + 5 \)[/tex]. So, the coefficients are [tex]\( 1 \)[/tex] and [tex]\( 5 \)[/tex].
3. Perform synthetic division:
- Step 1: Bring down the leading coefficient [tex]\( 1 \)[/tex].
- Step 2: Multiply this by the root [tex]\( 2 \)[/tex] and write the result under the next coefficient [tex]\( 5 \)[/tex].
- Step 3: Add the numbers in the second column: [tex]\( 5 + (2 \times 1) = 7 \)[/tex].
The setup looks like this:
```
2 | 1 5
| 2
------------
1 7
```
4. Interpret the result: After carrying out the division, the numbers at the bottom represent the coefficients of the quotient polynomial.
- The [tex]\( 1 \)[/tex] is the coefficient of [tex]\( x \)[/tex].
- The [tex]\( 7 \)[/tex] is the constant term.
5. Write the quotient in polynomial form: The result is [tex]\( x + 7 \)[/tex].
Therefore, the quotient is [tex]\( x + 7 \)[/tex].
Thus, the correct answer is [tex]\( \text{D. } x + 7 \)[/tex].
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