Multiple Root

Pronunciation: /ˈmʌl tə pəl rut/ ?

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+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 and 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)


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Manipulative 1: Multiple roots. Created with GeoGebra.

Manipulative 1 lets you change the multiplicity of the roots of the polynomial f(x)=(x-1)^m*(x+2)^n. Click on the green and blue points on the sliders and drag them to change the multiplicity.

  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?


  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. (Accessed: 2009-12-18).

Cite this article as:

Multiple Root. 2009-12-18. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2009-12-18: Added revision (McAdams, David.)
2008-12-16: Initial version (McAdams, David.)

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