If a polynomial function f(x) has roots at 4, 13i, and 5, then one of its factors is derived from these roots.
In polynomial functions, complex roots always occur in conjugate pairs. Since we have 13i as a root, its conjugate -13i must also be a root. Therefore, we can list the roots as 4, 5, 13i, and -13i.
To find the factors corresponding to these roots, we can express them in factor form. The factors related to these roots are:
- (x – 4)
- (x – 5)
- (x – 13i)
- (x + 13i)
From the given roots, we can see that at least one factor of f(x) must be (x – 4) and (x – 5), since those are the linear factors for the real roots. Moreover, to account for the complex roots, we combine the factors for 13i:
(x – 13i)(x + 13i) simplifies to (x^2 + 169).
Therefore, a polynomial function with these roots can be expressed in a factored form including all the roots:
f(x) = (x – 4)(x – 5)(x^2 + 169)
In conclusion, (x – 4) is a factor of f(x), and it is one of the expressions you could use to construct the complete polynomial.