Multiple Root

A multiple root of a polynomial is a root that occurs more than once.^[2] 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.

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?