Answer :
Sure! Let's go through the synthetic division process step-by-step.
We are asked to divide a polynomial using synthetic division. The polynomial given is related to a division problem in the format `2 \longdiv { 1 \quad 5 \quad -1 }`, which means we are dividing by the value `2` and using the coefficients `1`, `5`, and `-1`.
### Steps for Synthetic Division
1. Identify the divisor and the polynomial coefficients:
- Divisor (the number outside): `2`
- Coefficients of the polynomial (from left to right): `1` (for [tex]\(x^2\)[/tex]), `5` (for [tex]\(x\)[/tex]), `-1` (constant term).
2. Setup the process:
- Write the divisor `2` on the left.
- Write the coefficients [tex]\(1, 5, -1\)[/tex] from top to bottom on the right.
3. Bring down the leading coefficient:
- Start by bringing down the first coefficient `1` to the bottom row. We will call this a "carry."
4. Multiply and add recursively:
- Multiply the carry by the divisor: [tex]\(1 \times 2 = 2\)[/tex].
- Add this result to the next coefficient: [tex]\(5 + 2 = 7\)[/tex]. Write `7` beneath the line next to `1`.
5. Repeat the process:
- Multiply the new carry `7` by the divisor: [tex]\(7 \times 2 = 14\)[/tex].
- Add this result to the next coefficient: [tex]\(-1 + 14 = 13\)[/tex].
6. Interpret the result:
- The numbers at the bottom are the result of the division. The first two numbers, `1` and `7`, represent the coefficients of the quotient polynomial. The final number, `13`, is the remainder.
### Result in Polynomial Form
The coefficients `1` and `7` mean the quotient polynomial is:
[tex]\(1x + 7\)[/tex].
Thus, the quotient in polynomial form is:
A. [tex]\(x + 7\)[/tex]
The remainder is irrelevant for this particular question since it's not asked, but if needed, we'd mention it as `13`.
We are asked to divide a polynomial using synthetic division. The polynomial given is related to a division problem in the format `2 \longdiv { 1 \quad 5 \quad -1 }`, which means we are dividing by the value `2` and using the coefficients `1`, `5`, and `-1`.
### Steps for Synthetic Division
1. Identify the divisor and the polynomial coefficients:
- Divisor (the number outside): `2`
- Coefficients of the polynomial (from left to right): `1` (for [tex]\(x^2\)[/tex]), `5` (for [tex]\(x\)[/tex]), `-1` (constant term).
2. Setup the process:
- Write the divisor `2` on the left.
- Write the coefficients [tex]\(1, 5, -1\)[/tex] from top to bottom on the right.
3. Bring down the leading coefficient:
- Start by bringing down the first coefficient `1` to the bottom row. We will call this a "carry."
4. Multiply and add recursively:
- Multiply the carry by the divisor: [tex]\(1 \times 2 = 2\)[/tex].
- Add this result to the next coefficient: [tex]\(5 + 2 = 7\)[/tex]. Write `7` beneath the line next to `1`.
5. Repeat the process:
- Multiply the new carry `7` by the divisor: [tex]\(7 \times 2 = 14\)[/tex].
- Add this result to the next coefficient: [tex]\(-1 + 14 = 13\)[/tex].
6. Interpret the result:
- The numbers at the bottom are the result of the division. The first two numbers, `1` and `7`, represent the coefficients of the quotient polynomial. The final number, `13`, is the remainder.
### Result in Polynomial Form
The coefficients `1` and `7` mean the quotient polynomial is:
[tex]\(1x + 7\)[/tex].
Thus, the quotient in polynomial form is:
A. [tex]\(x + 7\)[/tex]
The remainder is irrelevant for this particular question since it's not asked, but if needed, we'd mention it as `13`.
