Provide an appropriate response.
-Suppose that $k , a , b$, and $c$ are real numbers and that the polynomial function $f$ may be expressed in factored form as $f ( x ) = ( x - k ) \left( a x ^ { 2 } + b x + c \right)$. If $b ^ { 2 } - 4 a c = - 57$, determine the number of real and imaginary zeros that $f$ has.
A) 3 real zeros, 0 imaginary zeros
B) 2 real zeros, 2 imaginary zeros
C) 2 real zeros, 1 imaginary zero
D) 1 real zero, 2 imaginary zeros

Question

Quizplus · Accepted Answer

The quadratic formula's discriminant, $b^2 - 4ac$, determines the nature of the roots of a quadratic equation $ax^2 + bx + c = 0$. If the discriminant is negative ($b^2 - 4ac < 0$), the equation has two imaginary (complex) roots. Since $b^2 - 4ac = -57$, the quadratic part of the polynomial $ax^2 + bx + c$ has two imaginary roots. The linear factor $(x - k)$ contributes one real root, making the total one real zero and two imaginary zeros.

Provide an Appropriate Response k,a,bk , a , bk,a,b , And ccc

Provide an Appropriate Response $k , a , b$ , And $c$