Question

what would be your first step in factoring the trinomial 6x^2-x-35

Answer 1

it depend of the method that u need to use
there r so many methods what u need to use?

Answer 2

Have you considered the modified form of the AC method? It is like the AC method, but not quite as tedious. I've found that for people without a developed factoring intuition, the following method can be a lifesaver. If we write quadratics like ax^2 + bx + c, then for your problem a = 6 and c = -35. Now, just like in the AC method, you multiply a times c ( 6 and -35) to arrive at -210. Now you are going to look for factors of -210 which also sum to -1 (the coefficient of your "b"). In practice, I usually drop the sign and just make 2 columns of factors (also without signs), filling in the signs "as needed" once I see a pair of factors that look like they could work. I also make it a point to list factors that I am nearly sure aren't the ones I need (for instance, 1 and 210, or 2 and 105, or 3 and 70, etc...) because I have been burned before by overlooking the correct factors (so I just list even the unlikely ones). By the time you arrive at 14 times 15, hopefully your intuition will suggest that with the correct signs, they seem like a likely pair. -15 + 14 does sum up to -1, which is what we were looking for.

Now you will just take those factors, and the coefficient of the x^2 term along with ONE of the x'es and set it up like so:

\[\frac{ 14 }{ 6x } and \frac{ -15 }{ 6x }\]

Now reduce both of those equations, arriving at:

\[\frac{ 7 }{ 3x } and \frac{ -5 }{ 2x }\]

Rearrange like so: (3x + 7) (2x - 5). You should always FOIL check your solutions, and if you do so you will find that these are the correct factors.

A word of caution: some math departments or teachers who aren't familiar with this method might initially discourage its use because the steps involve rewriting things that aren't mathematically equivalent (normally a big no-no). However, there are rather easy to follow proofs of this method easily found with an internet search. They may be beyond your current level (or interest), but a teacher or professor could easily understand them if they needed more convincing of the validity of this method and the rigor of the reasoning behind it.