Studying for math test - A few things I dont understand

[MadmanMurray]

I have a math test coming up and I have last years test in front of me for a study guide. I can do most of the questions on this test and I know the theory behind everything on it but here's a couple of questions that I can't seem to get right.

Q1) If (x + 1) and (x - 2) are factors of x³ + 2x² + ax + b find the values of a,b E(that greek symbol that looks like a rounded E) R, and find remaining factor.
I assumed that to find a and b I would have to plug each root -1 and 2 into the equation so I have 2 simultaneous equations but it didn't work out for some reason.

Heres what I did
(-1)³ + 2(-1)² + a(-1) + b = 0 =>a - b = 1
did the same with the other root and got
2a + b = 16

by elimination I got A=17/3 and B=5 but these don't seem to plug into the equation. Am I going about this question the right way?

Also it asks me to find the last factor. I know I can do that by dividing the factors I have into the equation but is that the quickest method to obtaining the final factor when you already have 2?

 

[NoMoreExams]

You can do it 2 ways,

1) you know that (x+1) and (x-2) are factors of that cubic, therefore you know the last root is real too (conjugate pairs and such) so it will be of the form (x - z)

So x^3 + 2x^2 + ax + b = (x+1)(x-2)(x-z), figure it out :)

2) since (x+1) and (x-2) are roots of that equation, then if we define

f(x) = x^3 + 2x^2 + ax + b, then f(-1) = f(2) = 0

Therefore -1 + 2 - a + b = 0 i.e. 1 - a + b = 0 which gives us a - b = 1 as you said

and

8 + 8 + 2a + b = 0 i.e. 16 + 2a + b = 0 which is the same as 2a + b = -16 which is not what you have. Maybe you want to try again?