Factoring x² - 24x - 17280 without trial and error

[bergausstein]

help factor this out using a faster and effective way.

$x^2-24x-17280$

I know that the factored form is $(x-144)(x+120)$

when I solved this I'm having a hard time finding a product of two numbers that will give me the middle term. can you give some fast way to determine that? with much less use of repetitious trial and error. thanks!

 

[MarkFL]

I would first look at the prime factorization:

$$17280=2^7\cdot3^3\cdot5$$

Now, we want to find two factors whose sum is $-24$, so we know the two factors will need to be close in value. So, we could try:

$$17280=\left(2^3\cdot3\cdot5\right)\left(2^3\cdot3\cdot6\right)=120\cdot144$$

And this turns out to be the factors we need. :D