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

1 Answer
Oct 26, 2016

Answer:

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.