The diagram shows a sketch of the graph of the curve y = x^3 - x together with t

Question

The diagram shows a sketch of the graph of the curve y = x^3 - x together with the tanget to the curve at the point
A(1, 0).
Use differentiation to find the equation of the tangent to the curve at A, and verify that the point B where the tangent
cuts the curve again has coordinates (-2, -6).
Use integration to find the area of the region bounded by the curve and the tangent (shaded in the diagram), giving
your answer as a fraction in its lowest terms. (Area is between curve and tangent line, mostly in bottom left quadrant).

Answers

a) y = 2x - 2

b) 6 3/4

I have worked out up to part (a) but cannot solve the cubic equation x^3- 3x + 2 =0

Explanation / Answer

To solve a cubic equation , you need to make a guess for 1 root and then divide it by that factor to obtain a quadratic expression. This Quadratic can then be calculated. in your Ques, x = 1 is one solution to the cubic, So x-1 is a factor of the equation.

Divide the cubic equation by x-1

gives = (x^2)+x -2, roots are = 1,-2

So i get x = 1 twice, what does that mean?

It means that at x = 1, the there is a local minima and it touches the x axis , so i get 2 roots repeated.

Solution = 1, -2

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