Multiplicity of Polynomial Roots

Question

Why is it important to consider multiplicity when determining the roots of a polynomial equation?

Answer 1

Multiplicity can tell us at least two things about a polynomial and its graph. First, multiplicity tells us the the number of repeating/repeatable factors a polynomial has to better determine the number of real (positive/negative) roots and complex roots a polynomial contains.

Example: (x+1)³(x- 4)²(x² + 4)

The above example has the following roots, x = -1 with a multiplicity of 3, x = 4, with a multiplicity of 2, and x = +2i & -2i, no multiplicity

Adding up the powers, tell us this polynomial has a degree ( highest power ) of 7. Since the polynomial is degree 7 and is odd, its behavior on the domain is (-∞, ∞) or down up or think statue of liberty as a visual metaphor.

Secondly, multiplicities tells us about the graph's behavior at the roots/x-intercepts. Roots with odd multiplicities cross over the x-axis from either a positive range to a negative range or vice versa. Even multiplicities, touch only the x-axis and bounce from, rather than cross the x-axis. So if the y or range values are positive and the curve of a graph approaches a root with even multiplicity, the curve will touch the x-intercept and return back into the positive y region of the graph, or if the y or range value is negative, it will return in the negative direction of y/range values on the graph.

In a nutshell, it can help us with graphing a polynomial as well as interpret a polynomial's curve behavior for the domain of all real numbers as well as the roots/x-intercepts based on its multiplicities.