Factoring 4th Powers

How would I factor a 4th power equation without guessing using long division?
ex. I LaPlaced a 4x4 matrix and ended up with:
x^4+16x^3+75x^2+108x
Where x's are lambda's. I'm solving for a matrix quadratic form (no cross-products), so I need to know the factors.

Question

Factoring 4th Powers


How would I factor a 4th power equation without guessing using long division? 
ex. I LaPlaced a 4x4 matrix and ended up with:
x^4+16x^3+75x^2+108x
Where x's are lambda's. I'm solving for a matrix quadratic form (no cross-products), so I need to know the factors.

anonymous · Answer

There is a quartic formula but for this polynomial you can pull a factor of x out of it to leave:
$$x(x^3+16x^2+75x+108) $$Then try factors of 109 (found -3 to be a factor), leaving us with:
$$x(x+3)(x^2+13x+36) $$Which we can factor easily:
$$x(x+3)(x+4)(x+9) $$Okay, I guessed a bit, but trying factors of 108 is a lot easier than the quartic formula!

anonymous · Answer

*factors of 108

anonymous · Answer

Ok thanks, I guess that's the best that I can get :)!

anonymous · Answer

No problem!