What is a good method to go about sketching a polynomial?

Let's do an example. Sketch: y=x^3 - 2x^2 - x + 2. First we need to find out where the curve crosses the x and y axes. Look at the x axis first. We factorise the expression if we can. The best method to factorise a cubic is to use the factor theorem. We substitute some small integers for x into the equation and see if the expression is equal to zero. Try x=1. 1^3 - 2(1^2) - 1 + 2 = 0. Therefore one of the factors of x^3 - 2x^2 - x + 2 is (x-1). Then we use algebraic division to find the other 2 factors. OR we could continue to use the factor theorem. The factor theorem is quicker so we will use that. Try x=-1. (-1)^3 - 2((-1)^2) - (-1) + 2 = 0.Therefore another one of the factors of x^3 - 2x^2 - x + 2 is (x+1). To find the third factor we can continue to guess or we can look at the constant term in x^3 - 2x^2 - x + 2, which is 2. Note that -112=2 since when we expand the brackets the constant term is the product of the negatives of the 3 roots. Therefore the final root is 2 so the final factor is (x-2). Therefore the factorised expression is y=(x-1)(x+1)(x-2) so the graph crosses the x axis at x= -1,1 and 2. The graph crosses the y axis when x=0. Substitute x=0 into y= x^3 - 2*x^2 - x + 2. We find the graph crosses the y axis at (0,2).Now for the general shape of the curve. The highest power of x is x^3. It is a positive power of x^3. Therefore we know the general shape (draw it). Now draw the graph connecting the points that we know and labelling where the graph crosses the x and y axes.