To determine a factor of the polynomial function f(x) given its roots, we start by recognizing the relationship between roots and factors of a polynomial. A root of a polynomial f(x) is a value of x that makes the polynomial equal to zero. If r is a root, then (x – r) is a factor of the polynomial.
In this case, we are provided with three roots: 4, 13i, and 5. This information allows us to write down the corresponding factors:
- For the root 4, the factor is (x – 4).
- For the root 5, the factor is (x – 5).
- For the root 13i, we must consider its complex conjugate. Since polynomials with real coefficients require complex roots to come in conjugate pairs, the conjugate of 13i, which is -13i, must also be a root. Thus, the factors associated with these complex roots are (x – 13i) and (x + 13i).
To summarize, the polynomial function f(x) has these factors:
- (x – 4)
- (x – 5)
- (x – 13i)
- (x + 13i)
Therefore, a possible expression for f(x) can be written as:
f(x) = k(x – 4)(x – 5)(x – 13i)(x + 13i), where k is a non-zero constant.
Conclusively, one of the necessary factors of f(x) must be (x – 4), (x – 5), or (x – 13i), accompanied by its conjugate (x + 13i).