how do i find all the roots/ factors to the equation- x^5 - 3x^4 - 2x^3 + 10x^2- 8?

Question

i know i have to find the boundaries....possibly do some synthetic division...can anyone help

Answer 1

Here is the factorization:
(x - 2) (x + 2) (x - sqrt(2)) (x + sqrt(2)) (x + 1)

So, how did I get that answer? -- Read on to see.

There is no algebraic method that can factor a general polynomial of order greater than 4. There are methods that work for specific cases, but there is no general algorithm.

Practically speaking, when you want to solve such a problem, there are two ways to go about it:
1. Try to find the factors through guesswork and trial-and-error. This often works if the polynomial is not too large, and if it has rational roots. (or at least roots that are expressible in a simple way)
2. Use a computer program to find the roots numerically. This always works. Once you have the roots, the factors are of the form (x - root).

I actually did it using method 2. Method 1 could have worked for this polynomial, but I was to lazy to do that.

Answer 2

General plan:

Start by testing all the integer factors of 8, probably in the sequence 1, -1, 2, -2, 4, ...

If any are roots, factor out the associated divisor x-r.

Hope that what's left over is manageable (in this case it is).