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Polynomial Graphs: Zeroes
     and Their Multiplicities (page 2 of 5)

Sections: End behavior, Zeroes and their multiplicities, "Flexing", Turnings & "bumps", Graphing

The real (that is, non-complex) zeroes of a polynomial correspond to the x-intercepts of the graph of that polynomial. So you can find the number of real zeroes of a polynomial by looking at the graph, and conversely you can tell how many times the graph is going to touch or cross the x-axis by looking at the zeroes of the polynomial (or the factored form of the polynomial).

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. For instance, the quadratic (x + 3)(x – 2) has the zeroes x = –3 and x = 2, each occuring once. The eleventh-degree polynomial (x + 3)⁴(x – 2)⁷ has the same zeroes, but in this case, x = –3 has multiplicity 4 and x = 2 has multiplicity 7, because of the number of times their factors occur.

The point of multiplicities with respect to graphing is that any factors that occur an even number of time (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:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

y = (x + 6)(x – 7)	y = (x + 6)(x – 7)2
x = –6 once x = 7 once	x = – 6 once x = 7 twice




y = (x + 6)2 (x – 7)	y = (x + 6)2(x – 7)2
x = – 6 twice x = 7 once	x = – 6 twice x = 7 twice

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 at that zero or turns back the way it came.

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

I can see from the graph that there are zeroes at x = –15, x = –10, x = –5, x = 0, x = 10, and x = 15, because the graph touches or crosses the x-axis at these points. (At least, I'm assuming that the graph crosses at exactly these points, since the problem doesn't tell me. When I'm guessing from a picture, I do have to make certain assumptions.) Since the graph just touches at x = –10 and x = 10, these zeroes occur an even number of times; the other zeroes occur an odd number of times. The odd-multiplicity zeroes might occur only once, or might occur three, five, or more times each; there is no way to tell from the graph (yet-- we'll learn more about that on the next page). And the even-multiplicity zeroes might occur four, six, or more times each; I can't tell by looking.

But if I add up the minimum multiplicity of each, I should end up with the degree, because otherwise this problem is asking for more information than is available for me to give. I've got the four odd-multiplicity zeroes (at x = –15, x = –5, x = 0, and x = 15) and the two even-multiplicity zeroes (at x = –10 and x = 10). Adding up their minimum multiplicities, I get 1 + 2 + 1 + 1 + 2 + 1 = 8, which is the degree of the polynomial. So the minimum multiplicities are the correct multiplicities.

x = –15 with multiplicity 1,
x = –10 with multiplicity 2,
x = –5 with multiplicity 1,
x = 0 with multiplicity 1,
x = 10 with multiplicity 2, and
x = 15 with multiplicity 1

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 numbers.

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