The graph of a polynomial function of degree 4 has no x-intercepts. describe the nature and number of all its zeros.
a) the function has no zeros.
b) the function has 4 real zeros.
c) the function has 4 imaginary zeros.
d) the function has 2 real and 2 imaginary zeros.

 = x^5 + 3x^3 − 4 

This function has at least one real root: 
x = 1 > y(1) = 0 

Since this is a 5th degree polynomial, it has at most 5 unique roots, and since we know at least 1 is real, then it must have at most 4 unique complex roots. 

Since the choices are 4, 5, 6, then answer can only be: 　4 

 

NOTE: x-intercepts are real zeros or roots 

The graphs of a polynomial function of degree 5 has three x-intercepts, all with multiplicity 1. 

A 5th degree polynomial will have 5 roots (counting multiplicities) 

Since this function has 3 x-intercepts, then it has 3 real roots, and 2 complex roots 

 

The polynomial function y = x^3 − 3x^2 + 16x − 48 has only one non-repeated x-intercept. What do you know about the complex zeros of the function? 

This is a 3rd degree polynomial, so it has 3 roots (counting multiplicities). 

Since it has only one non-repeated (i.e. multiplicity = 1) x-intercept, then it has 1 real root and 2 complex roots.