Final answer:
The graph of the odd-degree polynomial Q(x) = 2x³ + 5x² - 9x - 45 has a negative leading coefficient and an odd number of turning points. It does not have a local minimum or a positive leading coefficient.
Step-by-step explanation:
The function Q(x)=2x³+5x²−9x−45 is an odd-degree polynomial with a negative leading coefficient. The leading coefficient is the coefficient of the term with the highest degree, which in this case is -45. Since the degree of the polynomial is odd, the graph of Q(x) will have both ends pointing in the same direction. This means that the graph will start in the lower left quadrant and end in the upper right quadrant or vice versa.
Since the leading coefficient is negative (-45), the graph of Q(x) will start in the lower left quadrant and end in the upper right quadrant. Therefore, option d) The graph of Q(x) has a negative leading coefficient is true.
Option a) The graph of Q(x) has a local minimum is false because the degree of the polynomial is odd, and odd-degree polynomials do not have local minimum or maximum points. Option b) The graph of Q(x) has a positive leading coefficient is false because the leading coefficient is negative. Option c) The graph of Q(x) has an odd number of turning points is true because odd-degree polynomials have an odd number of turning points.