Step by step

This question is from Chapter 5. The purpose of this question is to have you develop the information required to give a rough sketch (by hand) of the polynomial function p(x) = 3x3 - 6x2 - 3x +6. Step 1. [2 marks]. State the degree of p(x). What does the degree and leading coefficient of p(I) tell us about the graph's end behavior? Use arrow notation to describe the end behavior. Step 2. [2 marks]. Use the Rational Zero Theorem to list all possible rational zeros (you do not need to test anything). Use Descartes' Rule of Signs to determine the possible number of positive and negative solutions. Step 3. [1 mark] Show that x = 2 is one of the rational zeros (you do not need to test any of the others from the list you created in the previous step). Step 4. [4 marks] Using the fact that X = 2 is a zero, use synthetic division to determine the remaining rational zeros. Write p(I) in factored form. Step 5. [3 marks]. From the previous step, write the T-intercepts as points. What are the multiplicities of the zeros and what does this tell us about the graph of p(i)? Step 6. [1 mark]. What is the y-intercept of p()? State your answer as a point. Step 7. [4 marks]. To help you create your graph, use the fact that there is a local minimum at the point (2+v7 20-14V7 this is approximately (1.549, -1.893) ], and a local maximum at the point (2 V7 20+14/7 3 ' 9 [this is approximately (-0.215, 6.338)]. Use this information (along with the end behavior from Step 1) to determine the intervals where the graph increases and decreases (write in interval notation). Step 8. [2 marks]. Use the information you determined in Steps 1 - 7 above to graph p(a). You cannot use any sort of graphing calculator for this, so just do a rough sketch by hand. Make sure you label all the information you found along the way (intercepts and local extrema)