To determine the polynomial function [tex]\( f(x) \)[/tex] with a leading coefficient of 1 and roots [tex]\( (7+i) \)[/tex] and [tex]\( (5-i) \)[/tex] each with multiplicity 1, we'll follow a logical sequence of steps:
1. Identify the complex conjugate pairs:
- The given roots are [tex]\( 7 + i \)[/tex] and [tex]\( 5 - i \)[/tex].
- For each given root, their complex conjugates are also roots of the polynomial if it has real coefficients.
- Thus, the roots we should consider are:
- [tex]\( 7 + i \)[/tex]
- [tex]\( 7 - i \)[/tex]
- [tex]\( 5 + i \)[/tex]
- [tex]\( 5 - i \)[/tex]
2. Write the polynomial in terms of its roots:
- If [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are the roots of the polynomial [tex]\( P(x) \)[/tex], the polynomial can be written as:
[tex]\[
P(x) = (x - a)(x - b)(x - c)(x - d)
\][/tex]
- Substituting the roots we have identified, we get:
[tex]\[
P(x) = (x - (7 + i))(x - (7 - i))(x - (5 + i))(x - (5 - i))
\][/tex]
3. Select the correct polynomial from the given options:
- The polynomial [tex]\( f(x) = (x-(7-i))(x-(5+i))(x-(7+i))(x-(5-i)) \)[/tex] matches the form determined above perfectly.
- This option correctly reflects all roots and the form [tex]\( (x - \text{root}) \)[/tex] for each of the roots.
4. Verification by comparison:
- We can double-check by a quick inspection to see:
- The roots match the required roots [tex]\( (7+i) \)[/tex] and [tex]\( (5-i) \)[/tex] and their conjugates [tex]\( (7-i) \)[/tex] and [tex]\( (5+i) \)[/tex].
- Other forms listed either do not have these specific roots or include the wrong roots or do not maintain the leading coefficient of 1.
Thus, the correct polynomial function is:
[tex]\[
f(x) = (x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))
\][/tex]
