which polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 - i) with multiplicity…

which polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 - i) with multiplicity 1?\no f(x)=(x + 7)(x - i)(x + 5)(x + i)\no f(x)=(x - 7)(x - i)(x - 5)(x + i)\no f(x)=(x-(7 - i))(x-(5 + i))(x-(7 + i))(x-(5 - i))\no f(x)=(x+(7 - i))(x+(5 + i))(x+(7 + i))(x+(5 - i))

which polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 - i) with multiplicity 1?\no f(x)=(x + 7)(x - i)(x + 5)(x + i)\no f(x)=(x - 7)(x - i)(x - 5)(x + i)\no f(x)=(x-(7 - i))(x-(5 + i))(x-(7 + i))(x-(5 - i))\no f(x)=(x+(7 - i))(x+(5 + i))(x+(7 + i))(x+(5 - i))

Answer

Explanation:

Step1: Recall complex - conjugate root theorem

If a polynomial with real coefficients has a complex root (a + bi), then its complex - conjugate (a - bi) is also a root. Given roots (7 + i) and (5 - i), the other roots are (7 - i) and (5 + i) since the polynomial has real coefficients.

Step2: Write the polynomial in factored form

For a root (r) of a polynomial, ((x - r)) is a factor of the polynomial. Since the roots are (7 + i), (7 - i), (5 + i), and (5 - i) and the leading coefficient is 1, the polynomial function is (f(x)=(x-(7 + i))(x-(7 - i))(x-(5 + i))(x-(5 - i))).

Answer:

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