Multiplicity is 2 because a cubic is only degree 3). This indicates that x 2 = 2 is a root of even multiplicity (in fact, the The graph does not cross the x-axis at the root x 2 = 2 (it simply touches the x-axis). Graph above, there are two distinct real roots, x 1 = −1 and x 2 = 2. The graph flattens out near this root because the root is not simple. Since the multiplicity is odd, the graph does cross the x-axis at the root, but The red graph above, there is one distinct real root, x = 0, having multiplicityģ. Of the root increases, the graph flattens out more and more near the root. Touch the x-axis at the root but will not cross the x-axis. If a root of a polynomial has even multiplicity, the graph will.If a root of a polynomial has odd multiplicity, the graph will cross the x-axis.The multiplicity of a root affects the shape of the graph of a polynomial. We will lookĪt the graphs of cubic functions with various combinations of roots and turning We will explore these ideas by lookingĬubic functions can have at most 3 real roots (including multiplicities) and 2 turning points. Most n real roots (including multiplicities) and n−1 turning points. Also recall that an n th degree polynomial can have at The limiting behavior of even and odd degree polynomials with positiveĪnd negative leading coefficients. In this section we will explore the graphs of polynomials. As usual, we use G − e to denote the graph obtained by deleting an edge e from a graph G and G/e the graph obtained by contracting e.Biology Project > Biomath > Polynomial Function > Polynomial Division Polynomial Functions Other recent progress has been made through the use of techniques from algebraic geometry to solve a long-standing open problem concerning the coefficients of chromatic polynomials.Īll graphs considered are finite and (unless stated otherwise) without loops or multiple edges. A combination of ideas and techniques from graph theory and statistical mechanics has led to significant new results on both polynomials. The chromatic polynomial is a specialization of the Potts model partition function, used by mathematical physicists to study phase transitions. Work on chromatic polynomials has received fresh impetus in recent years from an interaction with mathematical physics. Nevertheless, many beautiful results on chromatic polynomials have been obtained, and many other intriguing questions remain unanswered. This has not yet occurred: indeed, the four-colour conjecture is now a theorem, but the stronger conjecture of Birkhoff and Lewis remains open. Their hope was that results from analysis and algebra could be used to prove their stronger conjecture, and hence to deduce that the four-colour conjecture was true. Inspired by the four-colour conjecture, Birkhoff and Lewis obtained results concerning the distribution of the real zeros of chromatic polynomials of planar graphs and made the stronger conjecture that they have no real zeros greater than or equal to 4. The study of chromatic polynomials of graphs was initiated by Birkhoff in 1912 and continued by Whitney, in 1932. We also outline some techniques for working with these polynomials. We describe their basic properties and state some recent results and open problems. This chapter is concerned with chromatic polynomials and other related polynomials: flow polynomials, characteristic polynomials, Tutte polynomials and the Potts model partition function.
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