# Multiple Root

Pronunciation: /ˈmʌl.tə.pəl rut/ Explain 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. Take the polynomial in figure 1, (x - 1)(x - 1)(x + 2) The factor (x - 1) occurs twice, so it is a multiple root. Since it occurs only 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 (x - 1)(x - 1)(x - 1)(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. Figure 2: Graph of f(x) = (x - 1)(x - 1)(x - 1)(x + 2)

### Discovery

 Click on the points on the sliders and change them to change the figure. 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?
1. McAdams, David E.. All Math Words Dictionary, multiple root. 2nd Classroom edition 20150108-4799968. pg 120. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. 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 9/2/2018. http://www.gutenberg.org/files/29785/29785-pdf.pdf. Buy the book

McAdams, David E. Multiple Root. 4/25/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/m/multipleroot.html.

### Revision History

4/25/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)