Alg3.4.12
From Exampleproblems
This problem is a problem that is really tough to solve by standard methods, but if you notice a few things, then take a guess, then check the guess, it's not that bad.
The first thing to notice is that 8x3 and − 125 are both cubes. That puts me in the disposition to look to see if this whole monster is a perfect cube. If it is a cube, then it would be (2x − 5)3, since (2x)3 = 8x3 and ( − 5)3 = − 125. That's our guess, and so let's expand it using the binomial theorem, and see if if works.
Expanding, we get (2x − 5)3 = (2x)3 + (2x)2( − 5)(3) + (2x)( − 5)2(3) + ( − 5)3, which simplifies to what we want, 8x3 − 60x2 + 150x − 125.
--Spudbeach 23:53, 23 September 2007 (MDT)
