What is the area between y=x3 and its tangent at x=1
The Attempt at a Solution
The first derivative of y=x3, which is 3x2, tells me that the slope of the tangent at x=1 is 3. That (1,1) is a point on the tangent line tells me that the equation of the tangent line is y=3x-2. Now, I know that I need to solve y=3x-2 and y=x3 simultaneously to find the points of intersection, and that the x coordinates of these points will be my limits of integration. However, I'm having trouble factoring x3-3x+2=0. Anyone have any tips/tricks for factoring an equation like this? I suppose my question has more to do with algebra than calculus.