How do i completely factor this polynomial? (please explain)

[LostSong]

How do i completely factor this polynomial? (please explain)

2x²​+5x-12

 

[HallsofIvy]

If your question is "how can I look at this and immediately knowing how to factor this without trying different things?"- sorry, I can't answer that!
But I do know how you can get the factors with a little work.

There are a number of different ways to do it. If we insist upon integer coefficients, then it must be (2x+ something)(x+ something) since "2" can only be factored as 2 and 1. -12 is a bit harder! -1(12), (-2)(6), (-3)(4), (-4)(3), (-6)(2), and (-12)(1) all give -12.

So try them all!

(2x- 1)(x+ 12)= 2x^2- x+ 24x- 12= 2x^2+ 23x- 12. No, that doesn't work.

Keep going!

 

[Ishuda]

2x²​+5x-12

The first thing I do is, in part, as HallsofIvy said. You also have to have 2 times one of the factors plus the other factor add to 5. As an example, take what HallsofIvy did: 2 times the factor 12 minus 1 is 23, not 5 so that doesn't work (oh, and 2 times -1 plus 12 is 10 doesn't work for the other side of that particular factorization). That's a long list to work on so maybe there is another way. Well, yes there is. Any polynomial can be written as the product of its roots times the leading coefficient, so

2x²​+5x-12 = 2 (x-x₀) (x-x₁)

where x₀ and x₁ are solutions to

2x²​+5x-12 = 0.

Which is faster? Looks like maybe the roots method is faster. Then again, maybe not.

Had the equation been of higher order, say the highest order term involved x⁶, there are other methods one might try, i.e. the Rational Root Theorem
http://www.themathpage.com/aprecalc/factor-theorem.htm#strategy