Question 920515
the zeroes are -2, -1, and 3
this means the factors are (x+2), (x+1), and (x-3)


(x+2) * (x+1) = x^2 + x + 2x + 2
combine like terms to get x^2 + 3x + 2


x^2 + 3x + 2 * (x-3) = x^3 + 3x^2 + 2x - 3x^2 - 9x - 6
combine like terms to get x^3 - 7x - 6


if the equation goes through the point (2,10), then f(2) must be equal to 10.


f(2), however, is equal to 2^3 - 7*2 - 6 which is equal to 8 - 14 - 6 which is equal to -12.


you can't just add 22 to it because that will change the roots.


you can, however, multiply it by a factor that will allow f(2) to be equal to 10 and not change the roots.


that factor is -5/6.


how did we find it?


we set a * (x^3 - 7x - 6) = y


replace y with 10 and x with 2 and you get:


a * (2^3 - 7*2 - 6) = 10


simplify to get:


a * -12 = 10


divide both sides of the equation by -12 to get:


a = 10/-12.


simplify to get a = -5/6


your equation becomes:


-5/6 * (x^3 - 7x - 6) = y


simplify to get:


-5/6 * x^3 + 35/6 * x + 30/6 = y


this simplifies to:


-5/6 * x^3 + 35/6 * x + 5 = y


f(2) now becomes:


f(2) = -5/6 * 2^3 + 35/6 * 2 + 5 which becomes:


f(2) = -5/6 * 8 + 70/6 + 5 which becomes:


f(2) = -40/6 + 70/6 + 5 which becomes:


f(2) = 30/6 + 5 which becomes:


f(2) = 5 + 5 which is equal to 10.


your equation now satisfies the requirements and has the same roots as it had before.