Question

Write the equation of a polynomial function f(x) with real coefficients that has zeros 5i and 4, which has multiplicity 2.

a) f(x) = (x - 5i)(x + 5i)(x - 4)²
b) f(x) = (x - 5i)(x + 5i)(x - 4)
c) f(x) = (x + 5i)(x - 5i)(x - 4)²
d) f(x) = (x + 5i)(x - 5i)(x - 4)

Answer 1

The polynomial function with real coefficients that have zeros 5i and 4, with 4 having a multiplicity of 2, is f(x) = (x - 5i)(x + 5i)(x - 4)², which corresponds to option (a).

Step-by-step explanation:

The question asks to write the equation of a polynomial function f(x) with real coefficients that has zeros 5i and 4, and the zero at 4 should have a multiplicity of 2.

When a polynomial has complex zeros, they come in conjugate pairs, hence if 5i is a zero, -5i will also be a zero. Since the zero at 4 has a multiplicity of 2, it must be squared in the factorization. Therefore, taking all factors into account, the correct option is: f(x) = (x - 5i)(x + 5i)(x - 4)².So the correct answer is option (a).