To determine which polynomial function has a leading coefficient of 1 and roots [tex]\( (7+i) \)[/tex] and [tex]\( (5-i) \)[/tex] with multiplicity 1, we need to consider the properties of complex roots and polynomial construction.
### Step-by-Step Solution:
1. Identify the given roots and their conjugates:
- The given roots are [tex]\( 7+i \)[/tex] and [tex]\( 5-i \)[/tex].
- For polynomials with real coefficients, the complex root pairs must include their conjugates. Hence, the roots must also include [tex]\( 7-i \)[/tex] and [tex]\( 5+i \)[/tex].
2. Formulate the polynomial from the roots:
- The polynomial can be constructed by taking the roots and forming factors [tex]\((x - \text{root})\)[/tex]. Therefore, the factors are:
[tex]\[
(x - (7+i)), (x - (7-i)), (x - (5+i)), (x - (5-i))
\][/tex]
3. Construct the polynomial function:
- Combining these factors yields the polynomial:
[tex]\[
f(x) = (x - (7+i))(x - (7-i))(x - (5+i))(x - (5-i))
\][/tex]
4. Identify the correct polynomial from the given options:
- Let’s match this polynomial with the provided options:
- [tex]\( f(x) = (x + 7)(x - i)(x + 5)(x + i) \)[/tex]
- [tex]\( f(x) = (x - 7)(x - 1)(x - 5)(x + i) \)[/tex]
- [tex]\( f(x) = (x - (7-i))(x - (5+i))(x - (7+i))(x - (5-i)) \)[/tex]
- [tex]\( f(x) = (x + (7-i))(x + (5+i))(x + (7+i))(x + (5-i)) \)[/tex]
- Simplifying our identified polynomial expression:
[tex]\[
f(x) = (x-(7-i))(x - (7+i))(x - (5-i))(x - (5+i))
\][/tex]
- Looking closely at the provided options, the third option clearly matches:
[tex]\[
(x - (7-i))(x - (5+i))(x - (7+i))(x - (5-i))
\][/tex]
Thus, the correct polynomial function is:
[tex]\[
f(x) = (x-(7-i))(x-(5+i))(x-(7+i))(x-(5-i))
\][/tex]
The correct option is:
3
