Question 292578
Equation is:


-3t^3+23t^2+8t


Factor by "t" to get:


t * (-3t^2 + 23t + 8)


Set this equation equal to 0 to find the roots.


t * (-3t^2 + 23t + 8) = 0


Factor (-3t^2 + 23t + 8) to get:


t * (3t+1) * (-t+8) = 0


This makes t = 0 or (3t+1) = 0 or (-t+8) = 0


If 3t+1 = 0, then t = -1/3


If -t+8 = 0, then t = 8


or t = 0.


Your 3 possible roots are:


t = -1/3
t = 8
t = 0


Plug these into your original equation to see if they are good.


Your original equation is:


-3t^3+23t^2+8t


When t = 0, this equation becomes 0.


When t = -1/3, this equation becomes -3*(-1/3)^3 + 23*(-1/3)^2 + 8*(-1/3).


This becomes .11111111 + 2.55555555 - 2.66666666 which becomes 0.


When t = 8, this equation becomes -3*8^3 + 23*8^2 + 8*8.


This becomes -1536 + 1472 + 64 which becomes 0.


All 3 roots are good.


Our factored equation is:


N(t) = t * (3t+1) * (-t+8) = 0


When we substitute 3 for t, we get:


N(3) = 3 * (3*3+1) * (-3+8) which becomes:


N(3) = 3 * 10 * 5 which becomes:


N(3) = 150


Plug 3 into the original equation of -3t^3+23t^2+8t and you get -3*(3^3) + 23*(3^2) + 8*3 which becomes -3*27 + 23*9 + 24 which becomes -81 + 207 + 24 which becomes 150.


The factoring is good and the answer is N(3) = 150


I determined the factoring as follows:


I knew that 1*8 = 8 or 2*4 = 8 


I tried 1*8 first.


I knew that 3*8 = 24 and -1*8 = -1 so I figured that the factors had to be something like (3t+1) * (-t+8)


When you multiply these together, you get:


(3t+1) * (-t+8) =
(3t * (-t+8)) + (1 * (-t+8)) =
(-3t^2 + 24t) + (-t + 8) =
-3t^2 + 24t -t + 8 = 
-3t^2 + 23t + 8 which makes them the correct factors.