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Is it possible that the polynomial function doesn't have zeros?
In the real domain, yes. In the complex domain, no.
Can you have quadratic function with one real root and are complex root?
Yes, but in this case, the coefficients of the polynomial can not all be real.
What is the difference between a power functions and a polynomial functions?
A power function is of the form xa where a is a real number. A polynomial function is of the form anxn + an-1xn-1 + ... + a1x + a0 for some positive integer n, and all the ai are real constants.
When finding domain of a polynomial function is negative infinity to infinity the same as all real numbers?
Yes, that's the same thing.
Why the graph of a polynomial function with real coefficients must have a y-intercept but may have no x-intercept?
For a polynomial of the form y = p(x) (i.e., some polynomial function of x), having a y-intercept simply means that the polynomial is defined for x = 0 - and a polynomial is defined for any value of "x". As for the x-intercept: from left to right, a polynomial of even degree may come down, not quite reach zero, and then go back up again. A simple example is y = x2 + 1. Why is the situation for "x" and for "y" different? Well, the original equation is a polynomial in "x"; but if you solve for "x", you don't get a polynomial in "y".
