Multiple Root

Pronunciation: /ˈmʌl tə pəl rut/ Explain

Graph of f(x)=(x-1)(x-1)(x+2)
Figure 1: Graph of f(x)=(x-1)(x-1)(x+2)

A multiple root of a polynomial is a root that occurs more than once.[1] Take the polynomial in figure 1, f(x)=(x-1)(x-1)(x-1)(x+2)=(x-1)^3(x+2). The factor (x-1) occurs twice, so it is a multiple root. Since it occurs twice, it is also called a double root. The graph of polynomials with a double root just touches the x-axis at the root then changes direction.

Figure 2 shows the graph of the polynomial f(x)=(x-1)(x-1)(x-1)(x+2)=(x-1)^3(x+2). In this polynomial, the term (x-1) occurs 3 times. It has a multiplicity of 3. It is also called a triple root. Note that the graph of the polynomial crosses the x-axis at the triple root.

Graph of f(x)=(x-1)(x-1)(x-1)(x+2)
Figure 2: Graph of f(x)=(x-1)(x-1)(x-1)(x+2)

Discovery

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What can you say about m and n if the function is always positive, x always greater than zero?
Manipulative 1 - Multiple Root Created with GeoGebra.

  1. For what multiplicities (values of m and n) does the graph cross the axis at the root?
  2. For what multiplicities does the graph change direction at the root?
  3. What general rule can you make about the multiplicities of roots and whether the graph crosses the axis at the root?

References

  1. Dickson, Leonard Eugene, Ph.D.. First Course in the Theory of Equations. ch II pg 18. New York, John Wiley and Sons Inc.. 1922. Last Accessed 12/18/2009. http://www.gutenberg.org/files/29785/29785-pdf.pdf.

Cite this article as:

McAdams, David E. Multiple Root. 9/4/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/multipleroot.html.

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Revision History

9/4/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
12/18/2009: Added revision. (McAdams, David E.)
12/16/2008: Initial version. (McAdams, David E.)

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