To determine which polynomial function has a leading coefficient of 1 and roots [tex]\( (7+i) \)[/tex] and [tex]\( (5-i) \)[/tex] each with multiplicity 1, we will consider each of the given options.
### Roots and Polynomial Representation
Given roots: [tex]\( (7+i) \)[/tex] and [tex]\( (5-i) \)[/tex].
By the nature of polynomial equations and complex roots, any given polynomial with real coefficients must have its complex roots occur in conjugate pairs. Thus, the conjugates of the given roots are [tex]\( (7-i) \)[/tex] and [tex]\( (5+i) \)[/tex].
This highlights the suggested polynomial:
[tex]\[ (x - (7+i))(x - (7-i))(x - (5-i))(x - (5+i)) \][/tex]
Let's review each polynomial option to find the one that matches:
### Option 1:
[tex]\[ f(x) = (x + 7)(x - i)(x + 5)(x + i) \][/tex]
Check roots:
- [tex]\((x = 7+i)\)[/tex]: [tex]\((7+i +7)\)[/tex]
- Root not satisfied.
### Option 2:
[tex]\[ f(x) = (x - 7)(x - i)(x - 5)(x + i) \][/tex]
Check roots:
- [tex]\((x = 7+i)\)[/tex]: [tex]\((7+i -7)\)[/tex]
- Root not satisfied.
### Option 3:
[tex]\[ f(x) = (x - (7-i))(x - (5+i))(x - (7+i))(x - (5-i)) \][/tex]
Check roots:
- [tex]\( (x = 7+i)\)[/tex]: [tex]\((7+i - (7+i)) = 0 \)[/tex] (root satisfied)
- [tex]\( (x = 5-i)\)[/tex]: [tex]\((5-i - (5-i))= 0 \)[/tex] (root satisfied)
This polynomial form satisfies the conditions of the given roots.
### Option 4:
[tex]\[ f(x) = (x + (7-i))(x + (5+i))(x + (7+i))(x + (5-i)) \][/tex]
Check roots:
- [tex]\((x = 7+i\)[/tex]: [tex]\((7+i + (7-i))\)[/tex]
- Root not satisfied.
Therefore, the polynomial function that has a leading coefficient of 1 and roots [tex]\(7+i\)[/tex] and [tex]\(5-i\)[/tex] each with multiplicity 1 is:
[tex]\[ f(x) = (x-(7-i))(x-(5+i))(x-(7+i))(x-(5-i)). \][/tex]
Thus the correct answer is:
[tex]\[ \boxed{f(x) = (x-(7-i))(x-(5+i))(x-(7+i))(x-(5-i))} \][/tex]
