Could you please help me with the following:
>Cubic function f(x)=2x^3 + 6x^2 - 4.5x -13.5.
>1) Find the roots and confirm them by remainder theorem.
>2) Taking two roots at a time, find the equations of the tangent
>lines to the average of two of the three roots?
>3) Find where the tangent lines at the average of the two roots
>intersect the curve again.
>4) State a conjecture concerning the roots of the cubic and tangent
>lines at the average value of the roots. Proove it and
>investigate: one root, two roots and, one real and two complex roots.
I found the roots from the graph, (1.5,0), (-3,0) and (-1.5,0) but could some one tell me in steps how to factorise it to find the roots in the form
a(x-b)(x-c)(x-d). Or any other way of finding the roots.
When I'm supposed to proove it by remainder thorem, do I divide the function by one at a time, roots?
Then when I'm suppossed to find the tangent and I guess you take
((x-b)+(x-c))/2 for the average of roots, and i think you're suppposed to use differentiation to find it. Differentiate the function and then put the average of the roots in it, and if so what do you do with the number you get from this calculation? In what form should the tangent be written?
I'm also suppossed to see some pattern in my calculations (of 3. i guess)
For 3. am i supposed to look at each pair of roots, because I guess that it won't be all the pair that will intersect the curve again.