4.3.2 Real Zeros of Polynomial Functions - Part II
Multiplicity of a zero of a polynomial
Consider f(x) = 4x2 - x3
Factor it as much as possible: f(x) = x2(4 - x)
Note that
 f has zeros (touches or crosses x-axis) at
 0 (from the x2) and
 4 (from the (4 - x))
 we say that:
 0 is a zero of multiplicity 2, because the factor x has
2 as an exponent
 4 is a zero of multiplicity 1, because the factor
(4 - x) has 1 as an exponent
The multiplicity property
If a polynomial has a zero of:
 even multiplicity, it touches the x-axis at that zero, but
does not cross it
 odd multiplicity, it crosses the x axis at that zero
if the graph of a polynomial “flattens out” as it crosses the
x-axis, the zero is of multiplicity > 1 (3, 5, …)
4.3.2-1
Putting it all together
Consider again: f(x) = 4x2 - x3 = x2(4 - x).
 by the polynomial tail principal, the tails of f (leading
term -x3) rise on the left and fall on the right.
 its zeros are:
 0 (multiplicity of 2) - touches x-axis there
 4 (multiplicity 1) - crosses x-axis there
 just by plotting its zeros and noting their multiplicities,
we can create a rough graph of f:
left tail
right tail
Graph it on the graphing calculator, and see if this is
(roughly) correct.
4.3.2-2