State how many complex and real zeros the function has.
-$$f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2$$
A) 3 complex zeros; 2 real
B) 3 complex zeros; 1 real
C) 3 complex zeros; none real
D) 3 complex zeros; all 3 real

Question

Quizplus · Accepted Answer

The function is a cubic polynomial, which means it has exactly 3 complex zeros (including real zeros) by the Fundamental Theorem of Algebra. Since all coefficients are real, any complex zeros must come in conjugate pairs. However, a cubic polynomial cannot have exactly one pair of complex conjugate zeros (as that would imply it has four zeros in total, which is not possible). Therefore, all 3 zeros must be real.

State How Many Complex and Real Zeros the Function Has f(x)=x3−4x2+5x−2f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2f(x)=x3−4x2+5x−2

State How Many Complex and Real Zeros the Function Has $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2$