College

What is the remainder in the synthetic division problem below?

[tex]\[ 1 \longdiv 1\ 2\ -3\ 2 \][/tex]

A. 2
B. 4
C. 5

Answer :

To solve the synthetic division problem and find the remainder, let's go through the process step-by-step:

We are dividing the polynomial [tex]\(x^3 + 0x^2 - 3x + 2\)[/tex] by [tex]\(x - 1\)[/tex].

1. Identify the coefficients: The coefficients of the polynomial [tex]\(x^3 + 0x^2 - 3x + 2\)[/tex] are [tex]\([1, 0, -3, 2]\)[/tex].

2. Identify the divisor root: The divisor is [tex]\(x - 1\)[/tex], which means the root (value of [tex]\(x\)[/tex] that makes the divisor zero) is [tex]\(1\)[/tex].

3. Perform synthetic division:

- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply this by the root (1) and add to the next coefficient:
- [tex]\(1 \times 1 = 1\)[/tex], then [tex]\(0 + 1 = 1\)[/tex].
- Multiply the result by the root (1) and add to the next coefficient:
- [tex]\(1 \times 1 = 1\)[/tex], then [tex]\(-3 + 1 = -2\)[/tex].
- Multiply the result by the root (1) and add to the next coefficient:
- [tex]\(-2 \times 1 = -2\)[/tex], then [tex]\(2 - 2 = 0\)[/tex].

4. Determine the remainder: The last number at the bottom is the remainder.

In this case, after performing synthetic division, the remainder is [tex]\(0\)[/tex].

Therefore, the correct answer is:

None of the options (2, 4, or 5) are correct, since the remainder is actually 0. It seems there might be an error in the choices provided.