To determine the degree of the simplest polynomial with integer coefficients that has [tex]\(\sqrt{2}\)[/tex], 2, and [tex]\(2 + 2i\)[/tex] as some of its zeros, it is essential to also include the complex conjugates and pairs required by polynomials with real coefficients. Here's a detailed, step-by-step solution:
1. Identify the given zeros:
- [tex]\(\sqrt{2}\)[/tex]
- 2
- [tex]\(2 + 2i\)[/tex]
2. Include the complex conjugate:
- For any complex root [tex]\(a + bi\)[/tex], where [tex]\(b \neq 0\)[/tex], the complex conjugate [tex]\(a - bi\)[/tex] must also be a root if the polynomial has real coefficients.
- Thus, if [tex]\(2 + 2i\)[/tex] is a root, its complex conjugate [tex]\(2 - 2i\)[/tex] must also be a root.
3. Include the conjugate for the irrational root:
- For any irrational root, such as [tex]\(\sqrt{2}\)[/tex], its conjugate [tex]\(-\sqrt{2}\)[/tex] must also be a root to ensure that all coefficients of the polynomial are integers.
4. List all the necessary zeros:
- [tex]\(\sqrt{2}\)[/tex]
- [tex]\(-\sqrt{2}\)[/tex]
- 2
- [tex]\(2 + 2i\)[/tex]
- [tex]\(2 - 2i\)[/tex]
5. Count the total number of different zeros:
- [tex]\(\sqrt{2}\)[/tex]
- [tex]\(-\sqrt{2}\)[/tex]
- 2
- [tex]\(2 + 2i\)[/tex]
- [tex]\(2 - 2i\)[/tex]
6. Determine the degree of the polynomial:
- The polynomial must include all these five roots to ensure it satisfies the given conditions.
- Hence, a polynomial with these 5 zeros will be of at least degree 5.
Thus, the degree of the simplest polynomial with integer coefficients that has [tex]\(\sqrt{2}, 2\)[/tex], and [tex]\(2 + 2i\)[/tex] as some of its zeros must be 5.
The correct answer is: 5
