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Answer :
Sure! Let's solve the synthetic division problem step-by-step:
We're given a polynomial with coefficients [tex]\(1, 2, -3, 1\)[/tex] and we need to divide it by [tex]\(x + 2\)[/tex]. In synthetic division, using [tex]\(x + 2\)[/tex] means our divisor value, or root, is [tex]\(-2\)[/tex].
Here’s how we solve it:
1. Set up the synthetic division: Write the coefficients of the polynomial in a row: [tex]\(1, 2, -3, 1\)[/tex].
2. Write the divisor (root): This is [tex]\(-2\)[/tex].
3. Start with the left-most coefficient: Bring down the first coefficient ([tex]\(1\)[/tex]) as it is.
4. Multiply and add: Follow these steps for each column:
- Multiply the current number at the bottom by the divisor ([tex]\(-2\)[/tex]).
- Write the result under the next coefficient.
- Add this result to the next coefficient and write the sum below the line.
Here is the step-by-step breakdown:
- First column:
- Bring down [tex]\(1\)[/tex].
- Second column:
- Multiply [tex]\(1 \times (-2) = -2\)[/tex].
- Add to the next coefficient: [tex]\(2 + (-2) = 0\)[/tex].
- Third column:
- Multiply [tex]\(0 \times (-2) = 0\)[/tex].
- Add to the next coefficient: [tex]\(-3 + 0 = -3\)[/tex].
- Fourth column:
- Multiply [tex]\(-3 \times (-2) = 6\)[/tex].
- Add to the next coefficient: [tex]\(1 + 6 = 7\)[/tex].
5. Remainder: The last number you get is the remainder of the division. So, the remainder here is [tex]\(7\)[/tex].
The remainder in the synthetic division problem is [tex]\(7\)[/tex]. So, the correct answer is option D. 7.
We're given a polynomial with coefficients [tex]\(1, 2, -3, 1\)[/tex] and we need to divide it by [tex]\(x + 2\)[/tex]. In synthetic division, using [tex]\(x + 2\)[/tex] means our divisor value, or root, is [tex]\(-2\)[/tex].
Here’s how we solve it:
1. Set up the synthetic division: Write the coefficients of the polynomial in a row: [tex]\(1, 2, -3, 1\)[/tex].
2. Write the divisor (root): This is [tex]\(-2\)[/tex].
3. Start with the left-most coefficient: Bring down the first coefficient ([tex]\(1\)[/tex]) as it is.
4. Multiply and add: Follow these steps for each column:
- Multiply the current number at the bottom by the divisor ([tex]\(-2\)[/tex]).
- Write the result under the next coefficient.
- Add this result to the next coefficient and write the sum below the line.
Here is the step-by-step breakdown:
- First column:
- Bring down [tex]\(1\)[/tex].
- Second column:
- Multiply [tex]\(1 \times (-2) = -2\)[/tex].
- Add to the next coefficient: [tex]\(2 + (-2) = 0\)[/tex].
- Third column:
- Multiply [tex]\(0 \times (-2) = 0\)[/tex].
- Add to the next coefficient: [tex]\(-3 + 0 = -3\)[/tex].
- Fourth column:
- Multiply [tex]\(-3 \times (-2) = 6\)[/tex].
- Add to the next coefficient: [tex]\(1 + 6 = 7\)[/tex].
5. Remainder: The last number you get is the remainder of the division. So, the remainder here is [tex]\(7\)[/tex].
The remainder in the synthetic division problem is [tex]\(7\)[/tex]. So, the correct answer is option D. 7.
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