Final answer:
To graph the polynomial P(x)=x⁵+x⁴-7x³-x²+6x+3, use a graphing device to plot the points on the graph and identify the x- and y-intercepts, as well as the coordinates of the local extrema. The end behavior of the polynomial is as follows: as x approaches negative infinity, the values of P(x) approach negative infinity; as x approaches positive infinity, the values of P(x) approach positive infinity.
Step-by-step explanation:
To graph the polynomial P(x)=x⁵+x⁴-7x³-x²+6x+3, we can use a graphing device such as a graphing calculator or online graphing tool. Using the graphing device, we can plot the points on the graph and determine the x- and y-intercepts, as well as the coordinates of any local extrema.
To find the x-intercepts, we look for the values of x where the graph crosses the x-axis. These are the solutions to the equation P(x) = 0. To find the y-intercept, we substitute x = 0 into the equation and evaluate P(0). To find the coordinates of any local extrema, we look for the highest and lowest points on the graph.
The end behavior of the polynomial can be determined by looking at the leading term, which is x⁵. Since the degree of the polynomial is odd and the leading coefficient is positive, the end behavior is as follows: as x approaches negative infinity, the values of P(x) also approach negative infinity; as x approaches positive infinity, the values of P(x) also approach positive infinity.