Factor the polynomial t^2-t-2

gigi9

Someone plz help me to do this problem below. Thanks
1)a) Verify that y= e^(-x) and y= e^(2x)are both solutions of the differential equation y"-y'-2y=0
b) Factor the polynomial t^2-t-2. What connection do you see between the roots of this polynomial and the solutions of y"-y'-2y=0 ?---What does "roots" means in this case??..I don't understand.
*** t^2-t-2=(t-2)(t+1)

HallsofIvy

Homework Helper
Once again, this really should be in homework (its a territorial matter!) and you really should show what you have tried.

However I will give you a hint: To show that y= e-x and y= e2x are solutions to the differential equation y"-y'-2y=0 means to find y' and y" for each function, put them into the given equation and see what happens!

") Factor the polynomial t^2-t-2. What connection do you see between the roots of this polynomial and the solutions of y"-y'-2y=0 ?---What does "roots" means in this case??..I don't understand."

I started to write "exactly what it means in algebra" but you do have a point (perhaps inadverdant): Strictly speaking to talk about "roots" you must have equation, not just a polynomial. This problem should have said either "the roots of the equation t2- t- 2= 0" or "the zeroes of the polynomial t2-t-2".

Okay, you have factored t2- t- 2= (t+ 1)(t- 2). Now WHAT are the roots of t2- t- 2= (t+ 1)(t- 2)= 0 and what possible connection could that have with e-x and e2x?

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