Question

Is it possible for a degree `4` polynomial function to have one zero and its corresponding equation to have `4` roots? Explain.

Answer 1

Yes and no...

Explanation:

When we are counting roots or zeros, we may or may not include multiplicity in our count.

Consider for example, the quartic function:

`f(x) = x^4`

This function is zero for only one value of `x`, namely `x = 0`. So in one sense you could say that it has one zero.

The corresponding equation is:

`x^4 = 0`

By the Fundamental Theorem of Algebra, any quartic equation in one variable has exactly `4` roots - counting multiplicity. In this particular example, it has one root of multiplicity `4`, namely `x=0`.

There are reasons to count roots or zeros according to their multiplicity or not. It really depends on the context.

So we could say that `f(x) = x^4` has one zero or four and that `x^4=0` has one root or four.

Zeros always correspond to roots and vice versa, but the convention for each concerning multiplicity may vary.