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

Question

Quizplus · Accepted Answer

The Fundamental Theorem of Algebra states that a polynomial of degree n will have exactly n roots (zeros), which can be real or complex. Since $f(x)$ is a 4th degree polynomial, it must have 4 complex zeros (real numbers are also considered complex numbers). Without solving the polynomial, we cannot specify the exact number of real zeros, but it's rare for a polynomial of this form to have all zeros real or none real. Typically, real zeros come in pairs or are all real. Given the options and without specific calculation, the most plausible scenario is that it has 2 real zeros (and thus 2 complex ones that are not real), making option C the most likely correct choice without further calculation.

State How Many Complex and Real Zeros the Function Has f(x)=x4−4x3+5x2−4x+4f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 5 x ^ { 2 } - 4 x + 4f(x)=x4−4x3+5x2−4x+4

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