Question

Which of the following polynomial functions has a leading coefficient of 1 and roots (7 + i) and (5 - i) with multiplicity 1?

A) f(x)=(x-7-i)(x-5+1)
B) f(x)=(x+7-i)(x+5+i)
C) f(x)=(x-7-i)(x+5-i)
D) f(x)=(x+7+i)(x-5-i)

Answer 1

The polynomial function that has a leading coefficient of 1 and roots (7 + i) and (5 - i) with multiplicity 1 is f(x)=(x-7-i)(x+5-i).

Step-by-step explanation:

The polynomial function with a leading coefficient of 1 and roots (7 + i) and (5 - i) with multiplicity 1 is option C) f(x)=(x-7-i)(x+5-i).

When a complex number appears as a root, its conjugate will also be a root. The roots (7 + i) and (5 - i) have their conjugates as (7 - i) and (5 + i), respectively. So, the polynomial function will have the factors (x - (7 + i)), (x - (7 - i)), (x - (5 - i)), and (x - (5 + i)). When multiplied together, these factors simplify to option C.