Question

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

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

C: f(x) = (x – (7 – i))(x – (5 + i))(x – (7 + i))(x – (5 – i))

Explanation:

Answer 2

C.
`f(x)=(x-(7-i))(x-(5+i))(x-(7+i))(x-(5-i))`

Explanation:

We want to find the equation of a polynomial the following properties;

i. Leading coefficient is 1

ii. roots (7 + i) and (5 – i) with multiplicity 1

Recall the complex conjugate properties of the roots of a polynomial.

According to this property, if

`a+bi` is a root of a polynomial, then the complex conjugate,
`a-bi` is also a root.

This means that:

(7 - i) and (5 + i) with multiplicity 1 are also roots of this polynomial.

The complete set of roots are:

`x=(7+i),x=(7-i),x=(5-i),x=(5+i)`

Therefore the polynomial is:

`f(x)=(x-(7-i))(x-(5+i))(x-(7+i))(x-(5-i))`

The correct choice is C.