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?

 

[Architeuthis Dux]

I would recommend first double checking your calculations to ensure that you have the correct values for a and b. It is possible that there was an error in your calculations, which is why the values you obtained do not seem to work when plugged into the equation. If you are still unable to find the error, I would suggest seeking help from a teacher or tutor to review your approach and calculations.

As for finding the last factor, dividing the known factors into the equation is a valid method, but there may be other methods that could be quicker. I would recommend researching and exploring different techniques for finding the remaining factor, such as using the remainder theorem or the rational root theorem. It is important to have a good understanding of various techniques and strategies for problem solving in math, as it can help you approach different types of questions more efficiently.

In addition, I would suggest practicing with more practice problems to strengthen your understanding and application of the concepts. This will also help you identify any common errors you may be making and improve your problem solving skills.

Overall, it is important to approach math problems with a clear and logical mindset, double check your calculations, and seek help when needed. With practice and a strong understanding of the concepts, you will be able to tackle any math problem with confidence.