Chapter 5 - Polynomials and Polynomial Functions - 5-1 Polynomial Functions - Practice and Problem-Solving Exercises - Page 286: 48

Sign of leading coefficient: Positive Least Possible Degree: Three

RECALL: The end behavior of the graph of a polynomial function is dependent on its leading term $ax^{n}$ and its degree $n$. (i) If the leading term's degree is even: (a) the end behavior is up and up if the leading coefficient is positive and (b) the end behavior is down and down if the leading coefficient is negative. (ii) If the leading term's degree is odd: (a) the end behavior is down and up if the leading coefficient is positive and (b) the end behavior is up and down if the leading coefficient is negative. The end behavior is down, up. Therefore the sign of the leading coefficient is positive and the degree is odd. In general, the graph of a polynomial function of degree $n(n≥1)$ has at most $n−1 $ turning points. The graph of a polynomial function of odd degree has an even number of turning points. The graph of a polynomial function of even degree has an odd number of turning points. Degree 1: Zero turning points Degree 2: One turning point Degree 3: Zero or two turning points Since there are two turning points, the least possible degree is three. Therefore, the sign of the leading coefficient is positive and the least possible degree of the polynomial function is three.