Find the equation of the tangents and normal line at the points indicated y=x^3-7x^2+15x-9 at its points of intersection with the x-axis

Question

Can you help me answer this? Please?

Answer 1

You can almost guess what the roots have to be for a nice, easy solution, and that's what they are. We know that we have 1 or 3 real roots and, from the alternating signs, all are positive, and from the rational root theorem, they can be 1, 3, or 9. Just glancing at the equation tells us that 9 is too big so, since they must multiply to be 9, they are 1, 3, and, 3.

To show that this is the solution, do synthetic division with 1, and you will see that this is a root, and the other factor is (x^2 - 6x + 9), which is (x-3)^2, just like we guessed.

To get the tangent lines and normal lines, take the derivative and calculate. If the first derivative is not equal to 0, we can plug the first derivative m into y = mx + b for the tangent line. The normal line has slope -1/m. Plug that into the same type of equation - y = -1/m x + b. Of course, y will equal 0.

From the form of the function, 3 is a multiple root, so the derivative is 0 there. Thus, the tangent line is y = 0, and the normal line is x = 3. I leave it to you to solve at x = 1.