Note (Geometric implication of multiplicity of a root):
Let be a polynomial function. A real number is called a root of of multiplicityif divides but does not divide . If , we say is a simple root. The following conclusions can be drawn for a root of multiplicity :
(i)
If is even, then is tangent to the graph at . The graph does not cross
and is not a point of inflection (see exercise 2(i) ).
(ii)
If then crosses at at , and is not a tangent to . The point
may or may not be a point of inflection (see exercise 2(ii) ).
(iii)
If is odd and then the graph of crosses at and it is a point of inflection
be a polynomial function. A real number 
is called a 
of 
if 
divides 
but 
does not divide 
. If 
, we say 
is a simple root. The following conclusions can be drawn for a root 
of multiplicity 
: 
is even, then 
is tangent to the graph 
at 
. The graph 
does not cross 
and 
is not a point of inflection (see exercise 2(i) ). 
then 
crosses at 
at 
, and 
is not a tangent to 
. The point
may or may not be a point of inflection (see exercise 2(ii) ). 
is odd and 
then the graph of 
crosses 
at 
and it is a point of inflection 
