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

Please help me with this. Thank you.

Answer 1

Because this function is cubic, it may intersect with the x-axis no more than three times. When y = 0, a lucky guess for x is 1 which works indicating that the point (1,0) should be included. Because of this the factor (x-1) must divide into the polynomial equally. Doing so shows the other two factors to be (x-1) and (x-3). Since factor (x-3) appears twice, it must be double so the curve must touch x-axis at tangent. So the first answer is y=0 (or the x-axis). For the second, where x = 3, y'(1) = 3x^2 -14x + 15 = 3-14+15 = 4. Using y=mx+b and the point (1,0) gives you the equation of the second tangent line. To find normal lines use negative reciprocals for slopes and y= mx+b.