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

Oct 26, 2016

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:

$f \left(x\right) = {x}^{4}$

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 $f \left(x\right) = {x}^{4}$ 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.