Why is it important to consider multiplicity when determining the roots of a polynomial equation?

Answer 1

because every root counts.

Answer 2

let's look at an example,

x^3 - 3x + 2 = 0

this is a cubic, so it must have 3 roots, either real or complex
Because of the nice +2 at the end, let's try some values
(but only factors of 2)
let f(x) = x^3 - 3x + 2
f(1) = 1 - 3 + 2 = 0, surprise surprise, or was it luck? (of course I planned this)

then x-1 must be a factor and of course x = 1 is a root

by either long division or synthetic division you should be able to verify that
x^3 - 3x + 2 = 0
(x-1)(x^2 + x - 2) = 0
this factors further
(x-1)(x+2)(x-1) = 0
so x = 1, -2, 1

notice that x = 1 shows up twice, so that is considered a multiplicity of 2

When you a multiplicity of roots of an even number, the corresponding graph will touch the x-axis for that value of x
https://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E3+-+3x+%2B+2+

or y = (x+2)^3 (x-1)
will have roots of -2,-2,-2,1
with a multiplicity of 3 , roots are -2 and 1