Answer :
When we address a synthetic division problem, we're usually dealing with dividing a polynomial by a linear expression of the form [tex]\(x - c\)[/tex]. In the problem you presented [tex]\((1 \longdiv { 1 2 - 3 2 })\)[/tex], we need to identify the polynomial and the divisor.
Let's assume that the expression represents dividing a polynomial where the coefficients are given by 1, 2, -3, and 2. This implies that the polynomial is [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex]. Additionally, it looks like the divisor we are using is [tex]\(x - 1\)[/tex] because we have [tex]\(1\)[/tex] to the left of the division sign, which typically signifies [tex]\(x - 1\)[/tex].
To perform synthetic division:
1. Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 2\)[/tex].
2. Place the divisor's root (which is 1 for [tex]\(x - 1\)[/tex]) to the left.
3. Bring down the leading coefficient (1) below the line.
4. Multiply 1 by the root (1), and write the result under the next coefficient.
5. Add this result to the next coefficient: [tex]\(2 + 1 = 3\)[/tex]. Write this result below the line.
6. Repeat the multiplication and addition process:
- Multiply 3 by the root (1), and write it under the next coefficient: [tex]\(-3 + 3 = 0\)[/tex].
- Multiply 0 by the root (1), and write it under the last coefficient: [tex]\(2 + 0 = 2\)[/tex].
After following these steps, the last number you get is the remainder in the division process. In this case, the remainder is 2.
Therefore, the remainder when dividing the polynomial by [tex]\(x - 1\)[/tex] using synthetic division is [tex]\(\boxed{2}\)[/tex].
Let's assume that the expression represents dividing a polynomial where the coefficients are given by 1, 2, -3, and 2. This implies that the polynomial is [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex]. Additionally, it looks like the divisor we are using is [tex]\(x - 1\)[/tex] because we have [tex]\(1\)[/tex] to the left of the division sign, which typically signifies [tex]\(x - 1\)[/tex].
To perform synthetic division:
1. Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 2\)[/tex].
2. Place the divisor's root (which is 1 for [tex]\(x - 1\)[/tex]) to the left.
3. Bring down the leading coefficient (1) below the line.
4. Multiply 1 by the root (1), and write the result under the next coefficient.
5. Add this result to the next coefficient: [tex]\(2 + 1 = 3\)[/tex]. Write this result below the line.
6. Repeat the multiplication and addition process:
- Multiply 3 by the root (1), and write it under the next coefficient: [tex]\(-3 + 3 = 0\)[/tex].
- Multiply 0 by the root (1), and write it under the last coefficient: [tex]\(2 + 0 = 2\)[/tex].
After following these steps, the last number you get is the remainder in the division process. In this case, the remainder is 2.
Therefore, the remainder when dividing the polynomial by [tex]\(x - 1\)[/tex] using synthetic division is [tex]\(\boxed{2}\)[/tex].
