The Fundamental Theorem of Algebra states that a polynomial function of degree n has n zeros. However, when you graph a polynomial function of n degrees, you see fewer than n zeros. Assuming that the graph is zoomed out far enough to see all of the zeros, what could be the reason for this discrepancy?

Explanation:

There are two possible reasons for that discrepancy.

1. Complex zeroes.

The Fundamental Theorem of Algebra states that a polynomial function of degree n has n complex zeros, thus some of those zeros may be non real. Since the graph shows only the real zeros, then you will not see the complex zeros. For instance, when you graph the function f(x) = x² + 1, you will see that the graph does not show any zeros. That is because the two zeros are i and -i.

2. Multiplicity.

Some zeros may have multiplicity.

The zeros stated by the Fundamental Theorem of Algebra include multiplicity.

For instance, the graph of the function f(x)=x³ shows one zero which is (0,0).

If you write x³ as the product of three x's: x.x.x, you can see that each x represents a zero, but all of them are the same x = 0. Then you say the the function has root x = 0 with multiplicity 3. But the graph can only show (0,0) one time.

So, your roots for f(x) = x^2 are actually 0 (multiplicity 2).