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Answer :
Certainly! Let's solve the problem using synthetic division to find the remainder. We are given the problem setup:
[tex]\(1 \longdiv { 1 } 2 2 \quad -3 \quad 3\)[/tex]
Here's a step-by-step guide:
1. Identify the root from the divisor:
The divisor here seems to be of the form [tex]\(x - c\)[/tex]. It looks like we're dividing by [tex]\(x - 3\)[/tex], so the root we'll use in synthetic division is [tex]\(c = 3\)[/tex].
2. Set up the synthetic division table:
We write the coefficients of the polynomial in order from highest to lowest degree across the top: 1, 2, 2, -3.
3. Perform synthetic division:
- Start with the first coefficient: Bring down the leading coefficient, which is 1.
- Multiply and add:
- Multiply this number (1) by the root (3) and write the result under the next coefficient. So, [tex]\(1 \times 3 = 3\)[/tex].
- Add that result to the next coefficient: [tex]\(2 + 3 = 5\)[/tex].
- Repeat the multiply and add process for each subsequent step:
- Multiply [tex]\(5\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(15\)[/tex], and add to the next coefficient: [tex]\(2 + 15 = 17\)[/tex].
- Multiply [tex]\(17\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(51\)[/tex], and add to the last coefficient: [tex]\(-3 + 51 = 48\)[/tex].
4. Identify the remainder:
The last number in the row is the remainder of the division. In this case, the remainder is 48.
So, the remainder in the synthetic division problem is 48.
[tex]\(1 \longdiv { 1 } 2 2 \quad -3 \quad 3\)[/tex]
Here's a step-by-step guide:
1. Identify the root from the divisor:
The divisor here seems to be of the form [tex]\(x - c\)[/tex]. It looks like we're dividing by [tex]\(x - 3\)[/tex], so the root we'll use in synthetic division is [tex]\(c = 3\)[/tex].
2. Set up the synthetic division table:
We write the coefficients of the polynomial in order from highest to lowest degree across the top: 1, 2, 2, -3.
3. Perform synthetic division:
- Start with the first coefficient: Bring down the leading coefficient, which is 1.
- Multiply and add:
- Multiply this number (1) by the root (3) and write the result under the next coefficient. So, [tex]\(1 \times 3 = 3\)[/tex].
- Add that result to the next coefficient: [tex]\(2 + 3 = 5\)[/tex].
- Repeat the multiply and add process for each subsequent step:
- Multiply [tex]\(5\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(15\)[/tex], and add to the next coefficient: [tex]\(2 + 15 = 17\)[/tex].
- Multiply [tex]\(17\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(51\)[/tex], and add to the last coefficient: [tex]\(-3 + 51 = 48\)[/tex].
4. Identify the remainder:
The last number in the row is the remainder of the division. In this case, the remainder is 48.
So, the remainder in the synthetic division problem is 48.
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