Question

Move numbers into the table to indicate the number of nonreal roots for each of the given polynomials.

Response area with 3 blank spaces
Polynomial
Number of

Real Roots

Number of

Nonreal Roots

x Superscript 5 Baseline minus 2 x Superscript 4 Baseline plus 9 x Superscript 3 Baseline minus 18 x Superscript 2 Baseline 3
Blank space 1 empty

x plus 12 1
Blank space 2 empty

x Superscript 4 Baseline plus 5 0
Blank space 3 empty


Answer options with 6 options.

Answer 1

To determine the number of nonreal roots for each polynomial, we can analyze them one by one.

1. **Polynomial:** \(x^5 - 2x^4 + 9x^3 - 18x^2\)
- The polynomial has 5 roots in total (from degree 5).
- It has 3 real roots (as given).
- Therefore, the number of nonreal roots is \(5 - 3 = 2\).
- **Blank space 1:** 2

2. **Polynomial:** \(x + 12\)
- This is a linear polynomial and has exactly 1 real root.
- Hence, there are no nonreal roots.
- **Blank space 2:** 0

3. **Polynomial:** \(x^4 + 5\)
- This polynomial has degree 4, so it has 4 roots in total.
- Since \(x^4 + 5 = 0\) has no real solutions (the function \(x^4 + 5\) is always positive for real \(x\)), it has 0 real roots.
- Therefore, the number of nonreal roots is \(4 - 0 = 4\).
- **Blank space 3:** 4

Now we can fill in the blanks as follows:

- **Blank space 1:** 2
- **Blank space 2:** 0
- **Blank space 3:** 4

Final allocation:
- \(x^5 - 2x^4 + 9x^3 - 18x^2\): Nonreal roots = **2**
- \(x + 12\): Nonreal roots = **0**
- \(x^4 + 5\): Nonreal roots = **4**