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Answer :
To find the remainder when dividing a polynomial by using synthetic division, we follow a systematic process. The polynomial in this case is represented by the coefficients [tex]\([1, 2, -3, 1]\)[/tex] and we're dividing by [tex]\(x + 2\)[/tex], which means we use [tex]\(-2\)[/tex] for synthetic division.
Here are the steps for synthetic division:
1. Set up the problem: Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 1\)[/tex].
2. Bring down the leading coefficient: Start with the first coefficient, which is [tex]\(1\)[/tex].
3. Multiply and add:
- Multiply the first coefficient by the divisor, [tex]\(-2\)[/tex]. So, [tex]\(1 \times (-2) = -2\)[/tex].
- Add this result to the next coefficient: [tex]\(2 + (-2) = 0\)[/tex].
4. Repeat the process:
- Multiply the result [tex]\(0\)[/tex] by the divisor, [tex]\(-2\)[/tex]. So, [tex]\(0 \times (-2) = 0\)[/tex].
- Add this result to the next coefficient: [tex]\(-3 + 0 = -3\)[/tex].
5. Continue with multiplication and addition:
- Multiply [tex]\(-3\)[/tex] by [tex]\(-2\)[/tex]. So, [tex]\(-3 \times (-2) = 6\)[/tex].
- Add this result to the last coefficient: [tex]\(1 + 6 = 7\)[/tex].
At the end of the process, the final value, [tex]\(7\)[/tex], is the remainder of the division.
Therefore, the remainder when the polynomial [tex]\(1x^3 + 2x^2 - 3x + 1\)[/tex] is divided by [tex]\(x + 2\)[/tex] using synthetic division is:
C. 7
Here are the steps for synthetic division:
1. Set up the problem: Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 1\)[/tex].
2. Bring down the leading coefficient: Start with the first coefficient, which is [tex]\(1\)[/tex].
3. Multiply and add:
- Multiply the first coefficient by the divisor, [tex]\(-2\)[/tex]. So, [tex]\(1 \times (-2) = -2\)[/tex].
- Add this result to the next coefficient: [tex]\(2 + (-2) = 0\)[/tex].
4. Repeat the process:
- Multiply the result [tex]\(0\)[/tex] by the divisor, [tex]\(-2\)[/tex]. So, [tex]\(0 \times (-2) = 0\)[/tex].
- Add this result to the next coefficient: [tex]\(-3 + 0 = -3\)[/tex].
5. Continue with multiplication and addition:
- Multiply [tex]\(-3\)[/tex] by [tex]\(-2\)[/tex]. So, [tex]\(-3 \times (-2) = 6\)[/tex].
- Add this result to the last coefficient: [tex]\(1 + 6 = 7\)[/tex].
At the end of the process, the final value, [tex]\(7\)[/tex], is the remainder of the division.
Therefore, the remainder when the polynomial [tex]\(1x^3 + 2x^2 - 3x + 1\)[/tex] is divided by [tex]\(x + 2\)[/tex] using synthetic division is:
C. 7
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