Answer :
To solve the synthetic division problem, we're dividing the polynomial represented by the coefficients [tex]\([1, 5, -1, 4]\)[/tex] by [tex]\(x - 2\)[/tex]. Here's the step-by-step process:
1. Identify the divisor and the coefficients:
The problem gives us the divisor as [tex]\(x - 2\)[/tex], which means [tex]\(a = 2\)[/tex]. The coefficients of the polynomial are [tex]\([1, 5, -1, 4]\)[/tex].
2. Set up the synthetic division:
- Write down the coefficients: 1, 5, -1, 4.
- Place the value 2 (from [tex]\(x - 2\)[/tex]) to the left.
3. Begin synthetic division:
- Bring down the leading coefficient (1) to start the new row.
4. Compute each step by multiplying and adding:
- Multiply 2 by the number just brought down (1), then add this result to the next coefficient (5):
[tex]\[
2 \times 1 = 2 \quad \text{and} \quad 5 + 2 = 7
\][/tex]
- Now, multiply 2 by 7 (result of the previous step), and add to the next coefficient (-1):
[tex]\[
2 \times 7 = 14 \quad \text{and} \quad -1 + 14 = 13
\][/tex]
- Multiply 2 by 13 (result of the previous step), and add to the last coefficient (4):
[tex]\[
2 \times 13 = 26 \quad \text{and} \quad 4 + 26 = 30
\][/tex]
5. Identify the quotient and remainder:
- The numbers below the line (except the last one) are the coefficients of the quotient. In this case, the quotient is represented by:
[tex]\[
1 \, x + 7
\][/tex]
- The last number (30) is the remainder.
6. Write the quotient in polynomial form:
The quotient, when written in polynomial form, is [tex]\(x + 7\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{x + 7}\)[/tex], which corresponds to option D in the choices given.
1. Identify the divisor and the coefficients:
The problem gives us the divisor as [tex]\(x - 2\)[/tex], which means [tex]\(a = 2\)[/tex]. The coefficients of the polynomial are [tex]\([1, 5, -1, 4]\)[/tex].
2. Set up the synthetic division:
- Write down the coefficients: 1, 5, -1, 4.
- Place the value 2 (from [tex]\(x - 2\)[/tex]) to the left.
3. Begin synthetic division:
- Bring down the leading coefficient (1) to start the new row.
4. Compute each step by multiplying and adding:
- Multiply 2 by the number just brought down (1), then add this result to the next coefficient (5):
[tex]\[
2 \times 1 = 2 \quad \text{and} \quad 5 + 2 = 7
\][/tex]
- Now, multiply 2 by 7 (result of the previous step), and add to the next coefficient (-1):
[tex]\[
2 \times 7 = 14 \quad \text{and} \quad -1 + 14 = 13
\][/tex]
- Multiply 2 by 13 (result of the previous step), and add to the last coefficient (4):
[tex]\[
2 \times 13 = 26 \quad \text{and} \quad 4 + 26 = 30
\][/tex]
5. Identify the quotient and remainder:
- The numbers below the line (except the last one) are the coefficients of the quotient. In this case, the quotient is represented by:
[tex]\[
1 \, x + 7
\][/tex]
- The last number (30) is the remainder.
6. Write the quotient in polynomial form:
The quotient, when written in polynomial form, is [tex]\(x + 7\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{x + 7}\)[/tex], which corresponds to option D in the choices given.
