The x-intercept
x
=
−
3
is the solution to the equation
(
x
+
3
)
=
0
. The graph passes directly through the x-intercept at
x
=
−
3
. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The x-intercept
x
=
2
is the repeated solution to the equation
(
x
−
2
)
2
=
0
. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.
(
x
−
2
)
2
=
(
x
−
2
)
(
x
−
2
)
The factor is repeated, that is, the factor
(
x
−
2
)
appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor,
x
=
2
, has multiplicity 2 because the factor
(
x
−
2
)
occurs twice.
The x-intercept
x
=
−
1
is the repeated solution of factor
(
x
+
1
)
3
=
0
. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function
f
(
x
)
=
x
3
. We call this a triple zero, or a zero with multiplicity 3.
For zeros with even multiplicities, the graphs touch or are tangent to the x-axis at these x-values. For zeros with odd multiplicities, the graphs cross or intersect the x-axis at these x-values. See the graphs below for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the x-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the x-axis.
For higher odd powers, such as 5, 7, and 9, the graph will still cross through the x-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis.
