math
posted by rita on .
find the coordinates of the point where the tangent to the curve y=x^3 +x +2 at the point (1,4) meets the curve again. [ans:2,8]
pls help me i don't understand the question....

dy/dx = 3x^2 + 1
at (1,4) , dy/dx = 4
tangent equation
4 = 4(1) + b
b = 0
the tangent equation at (1,4) is y = 4x
where does this cross or touch y = x^3 + x + 2 ?
x^3 + x + 2 = 4x
x^3  3x + 2 = 0
let f(x) = x^3  3x + 2
f(1) = 1  3 + 2 = 0 , so x1) is a factor
by synthetic division ...
x^3  3x + 2 = (x1)(x^2 + x  2)
= (x1)(x1)(x+2)
then the roots of x^3  3x + 2 = 0 are
x = 1 and x = 1 and x = 2
we knew x = 1 from the tangent part, so the new x = 2 is the other intersection point value
if x = 2
y = 8  2 + 2 = 8
the other point is (2,8)