Which polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 – i) with multiplicity 1?

Question

a. f(x) = (x + 7)(x – i)(x + 5)(x + i)
b. f(x) = (x – 7)(x – i)(x – 5)(x + i)
c. f(x) = (x – (7 – i))(x – (5 + i))(x – (7 + i))(x – (5 – i))
d. f(x) = (x + (7 – i))(x + (5 + i))(x + (7 + i))(x + (5 – i))

Answer 1

Since 7+i and 5-i are both roots and are both complex numbers, their complex conjugates 7-i and 5+i are also roots.

 

So you have 4 roots, hence 4 factors:

 

x-(7+i)

x-(7-i)

x-(5-i)

x-(5+i)

 

Multiply these 4 factors, leaving a 1 in front of the product:

 

Answer (c) has the correct factors:

 

f(x) = [x-(7-i)] [x-(5+i)] [x-(7+i)] [ x-(5-i)]