Question

If the degree of a polynomial is 4, then which of the following is always true?Choose the correct answer below.A.It crosses the x-axis 4 times.B.It crosses the x-axis at the origin.C.It will have exactly 4 real zeros.D.It will have exactly 4 complex zeros.

Answer 1

A polynomial of degree 4 will have exactly 4 complex zeros, which includes real and non-real numbers, but does not necessarily cross the x-axis 4 times or at the origin.

Step-by-step explanation:

If the degree of a polynomial is 4, this means that the highest power of the variable in polynomial terms is 4. The Fundamental Theorem of Algebra tells us that a polynomial of degree n will have exactly n complex zeros (which include both real zeros and non-real complex zeros). Therefore, a polynomial of degree 4 will have exactly 4 complex zeros. However, this does not guarantee how many times the polynomial will cross the x-axis, as complex zeros may not be real numbers, and real zeros may have multiplicities greater than 1. Moreover, the polynomial does not necessarily cross the x-axis at the origin unless one of its zeros happens to be zero.