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Answer :
Let's solve the synthetic division problem step-by-step to find the remainder.
We have a polynomial represented by its coefficients: [tex]\(1x^3 + 2x^2 - 3x + 1\)[/tex], and we are dividing it by [tex]\(x + 2\)[/tex], which means we're using [tex]\(x = -2\)[/tex] for synthetic division.
1. Set up the synthetic division:
- List the coefficients of the polynomial: [tex]\(1, 2, -3, 1\)[/tex].
- Write the divisor value [tex]\(-2\)[/tex] outside the division bracket.
2. Bring down the leading coefficient:
- The first coefficient is [tex]\(1\)[/tex]. Bring it straight down to the bottom row. This will be the starting point for calculations.
3. Multiply and add:
- Multiply the value you just brought down by [tex]\(-2\)[/tex]. So, [tex]\(1 \times -2 = -2\)[/tex].
- Add this result to the next coefficient: [tex]\(2 + (-2) = 0\)[/tex].
- Bring down the result ([tex]\(0\)[/tex]).
4. Repeat the multiply and add process:
- Multiply the new bottom number ([tex]\(0\)[/tex]) by [tex]\(-2\)[/tex]: [tex]\(0 \times -2 = 0\)[/tex].
- Add this to the next coefficient: [tex]\(-3 + 0 = -3\)[/tex].
- Bring down the result ([tex]\(-3\)[/tex]).
5. Continue the process:
- Multiply the new bottom number ([tex]\(-3\)[/tex]) by [tex]\(-2\)[/tex]: [tex]\(-3 \times -2 = 6\)[/tex].
- Add this to the last coefficient: [tex]\(1 + 6 = 7\)[/tex].
- Bring down the result ([tex]\(7\)[/tex]).
The final number you bring down at the end of the operation, which is [tex]\(7\)[/tex], is the remainder when dividing the polynomial by [tex]\(x + 2\)[/tex].
Thus, the remainder of the synthetic division is 7. The correct answer is B.
We have a polynomial represented by its coefficients: [tex]\(1x^3 + 2x^2 - 3x + 1\)[/tex], and we are dividing it by [tex]\(x + 2\)[/tex], which means we're using [tex]\(x = -2\)[/tex] for synthetic division.
1. Set up the synthetic division:
- List the coefficients of the polynomial: [tex]\(1, 2, -3, 1\)[/tex].
- Write the divisor value [tex]\(-2\)[/tex] outside the division bracket.
2. Bring down the leading coefficient:
- The first coefficient is [tex]\(1\)[/tex]. Bring it straight down to the bottom row. This will be the starting point for calculations.
3. Multiply and add:
- Multiply the value you just brought down by [tex]\(-2\)[/tex]. So, [tex]\(1 \times -2 = -2\)[/tex].
- Add this result to the next coefficient: [tex]\(2 + (-2) = 0\)[/tex].
- Bring down the result ([tex]\(0\)[/tex]).
4. Repeat the multiply and add process:
- Multiply the new bottom number ([tex]\(0\)[/tex]) by [tex]\(-2\)[/tex]: [tex]\(0 \times -2 = 0\)[/tex].
- Add this to the next coefficient: [tex]\(-3 + 0 = -3\)[/tex].
- Bring down the result ([tex]\(-3\)[/tex]).
5. Continue the process:
- Multiply the new bottom number ([tex]\(-3\)[/tex]) by [tex]\(-2\)[/tex]: [tex]\(-3 \times -2 = 6\)[/tex].
- Add this to the last coefficient: [tex]\(1 + 6 = 7\)[/tex].
- Bring down the result ([tex]\(7\)[/tex]).
The final number you bring down at the end of the operation, which is [tex]\(7\)[/tex], is the remainder when dividing the polynomial by [tex]\(x + 2\)[/tex].
Thus, the remainder of the synthetic division is 7. The correct answer is B.
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