Factorize Problem Homework: x^3 - 2x^2 -15x +36

[cathal84]

Homework Statement

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

Homework Equations

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

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

 

[fresh_42]

You perform a long division, which is simply analog to the way ordinary divisions are performed. In this case
##(x^3-2x^2-15x+36) : (x-3) = x^2 + ...##

I once typed in an example:
https://www.physicsforums.com/threa...r-a-polynomial-over-z-z3.889140/#post-5595083

It's a bit difficult to do it here without spoken explanations. Maybe it helps anyway.

 

[Ray Vickson]

Homework Statement

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

Homework Equations

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

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

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.

 

[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 :)

 

[Ray Vickson]

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 :)

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)##.)