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?\n○ f(x)=(x + 7)(x - i)(x + 5)(x + i)\n○ f(x)=(x - 7)(x - i)(x - 5)(x + i)\n○ f(x)=(x-(7 - i))(x-(5 + i))(x-(7 + i))(x-(5 - i))\n○ 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?\n○ f(x)=(x + 7)(x - i)(x + 5)(x + i)\n○ f(x)=(x - 7)(x - i)(x - 5)(x + i)\n○ f(x)=(x-(7 - i))(x-(5 + i))(x-(7 + i))(x-(5 - i))\n○ f(x)=(x+(7 - i))(x+(5 + i))(x+(7 + i))(x+(5 - i))

Answer

Explanation:

Step1: Recall root - polynomial relationship

If (r) is a root of a polynomial, then ((x - r)) is a factor of the polynomial. Given roots (7 + i) and (5 - i) with multiplicity 1, the factors of the polynomial are ((x-(7 + i))) and ((x-(5 - i))). Also, for a polynomial with real - valued coefficients, if (a+bi) is a root, then its complex conjugate (a - bi) is also a root. The complex conjugate of (7 + i) is (7 - i) and the complex conjugate of (5 - i) is (5 + i). So the factors of the polynomial are ((x-(7 - i))), ((x-(5 + i))), ((x-(7 + i))) and ((x-(5 - i))).

Step2: Write the polynomial function

The polynomial function (f(x)) with leading coefficient 1 and the given roots is (f(x)=(x-(7 - i))(x-(5 + i))(x-(7 + i))(x-(5 - i))).

Answer:

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