High School

What is the remainder in the synthetic division problem below?

[tex]\sqrt{12-32}[/tex]

A. 5
B. 4
C. 3
D. 2

Answer :

To solve the problem, let's focus on evaluating the expression under the square root: [tex]\(\sqrt{12-32}\)[/tex].

1. Calculate the Expression Inside the Square Root:
- Subtract 32 from 12:
[tex]\[
12 - 32 = -20
\][/tex]

2. Evaluate the Square Root:
- The expression under the square root is [tex]\(-20\)[/tex], which is a negative number.
- In real numbers, the square root of a negative number is not defined because no real number squared gives a negative result.

3. Complex Numbers:
- If we consider complex numbers, the square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i^2 = -1\)[/tex].

Since the square root involves a complex number, a real number remainder does not exist from this mathematical operation. Therefore, none of the options provided (A, B, C, D) are applicable as the problem involves a complex number situation.

The key takeaway is that the expression has no real remainder due to the presence of a negative number under the square root.