Given the roots of the polynomial function are [tex]\(2+i\)[/tex] and 5, let's find other possible roots and write the equation for this polynomial.
### Roots Analysis
1. Given Complex Root (2+i):
- For polynomials with real coefficients, complex roots occur in conjugate pairs.
- Therefore, the complex conjugate root [tex]\(2-i\)[/tex] must also be a root.
2. Given Real Root (5):
- This root is already provided in the problem statement.
### Conclusion on Roots
With the roots [tex]\(2+i\)[/tex], [tex]\(2-i\)[/tex], and 5 identified, we can write the polynomial equation using these roots.
### Polynomial Equation
The polynomial can be expressed as a product of factors derived from its roots:
[tex]\[ f(x) = (x - (2+i))(x - (2-i))(x - 5) \][/tex]
Now, let's examine the choices for the polynomial function and see which one matches our result:
1. [tex]\( f(x) = (x + (2+i))(x + (2-i))(x + 5) \)[/tex]
- This is incorrect because the signs in the factors do not match the roots.
2. [tex]\( f(x) = (x - (2+i))(x - (2-i))(x - 5) \)[/tex]
- This is correct as the signs correspond to the subtraction of each root from [tex]\(x\)[/tex].
3. [tex]\( f(x) = (x - (2+i))(x - (2-i)) \)[/tex]
- This is incorrect because it does not include the root [tex]\(5\)[/tex].
4. [tex]\( f(x) = (x + (2+i))(x - (2-i))(x - 5) \)[/tex]
- This is incorrect because it has an incorrect sign for one of the factors.
### Additional Root
Given the choices for additional roots:
- [tex]\(-3\)[/tex]
- [tex]\(-5\)[/tex]
- [tex]\(2-i\)[/tex]
- [tex]\(2i\)[/tex]
The required additional root for a polynomial with real coefficients is [tex]\(2-i\)[/tex]. This is the complex conjugate pair for [tex]\(2+i\)[/tex].
### Final Polynomial Function
Thus, the correct polynomial function is:
[tex]\[ f(x) = (x - (2+i))(x - (2-i))(x - 5). \][/tex]
To sum up:
- Additional Root: [tex]\(2-i\)[/tex]
- Equation of the Polynomial Function: [tex]\( f(x) = (x - (2+i))(x - (2-i))(x - 5) \)[/tex].
