Zeroes and Their Multiplicities

• List the zeroes, with their multiplicities, of the polynomial function y = 3(x + 5)³ (x + 2)⁴ (x − 1)² (x − 5)

---

The point of multiplicities with respect to graphing is that any factors that occur an even number of times (that is, any zeroes that occur twice, four times, six times, etc) are squares, so they don't change sign. Squares are always positive. This means that the x-intercept corresponding to an even-multiplicity zero can't cross the x-axis, because the zero can't cause the graph to change sign from positive (above the x-axis) to negative (below the x-axis), or vice versa.

The practical upshot is that an even-multiplicity zero makes the graph just barely touch the x-axis, and then turns it back around the way it came. You can see this in the following graphs:

All four graphs have the same zeroes, at x = −6 and at x = 7, but the multiplicity of the zero determines whether the graph crosses the x-axis at that zero or if it instead turns back the way it came. If the zero was of multiplicity 1, the graph crossed the x-axis at the zero; if the zero was of multiplicity 2, the graph just "kissed" the x-axis before heading back the way it came.

---

• The following graph shows an eighth-degree polynomial. List the polynomial's zeroes with their multiplicities.

---

I was able to compute the multiplicities of the zeroes in part from the fact that the multiplicities will add up to the degree of the polynomial, or two less, or four less, etc, depending on how many complex zeroes there might be. But multiplicity problems don't usually get into complex-valued roots. So, if you're asked to guess multiplicities from a graph, as above, you're probably safe in assuming that all of the roots are real numbers.