Factorizing Problem

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1. May 1, 2017

cathal84

1. The problem statement, all variables and given/known data
Hello, pretty back to basics with this one. How this came about was I am finding the eigenvalues for a given matrix and after forming the characteristic polynomial of the matrix i get this.

x^3 - 2x^2 -15x +36

2. Relevant equations
Using the rational root theorem i came to the conclusion that i have factor out (x-3) out of my function above.

3. The attempt at a solution
The answer after you factor out (x-3) from x^3 - 2x^2 -15x +36
is equal to (x-3)(x^2+x-12)
I have confirmation that this is the correct answer but i do not understand how to get there.
So if someone could show me how it is done step by step that would be great. Thanks

2. May 1, 2017

3. May 1, 2017

Ray Vickson

PF rules forbid us from showing you solutions step-by-step; we can give hints only.

Anyway, the standard way to do such tasks is to divide out the known factor (x-3) and then deal with the resulting quadratic. That involves "long division", I'm afraid.

4. May 1, 2017

cathal84

Alright thanks guys for letting me know it was long division anyway least i know what has to be done now! ill try figure it out myself :)

5. May 1, 2017

Ray Vickson

Another way is to express $p(x) = x^3 - 2 x^2 - 15 x+36$ as $p(x) = (x-3)(x^2+ax+b)$, and to expand the latter out. The resulting coefficients of $x^2, x$ and 1 will be expressions involving $a$ and $b$. Equating those expressions to -2, -15 and +36 will tell you what must be the values of $a$ and $b$. (In fact, you have three equations in the two unknowns $a$ and $b$, but they are consistent because $x = 3$ is an exact root of $p(x)$.)