Well done. I am a long since accomplished musician and have known these things for some time. Just as one of your other readers said about knowing the whole thing, I must agree. I have been struggling with Precalculus and its seem to have to do with the inability to switch over my problem solving skills with the guitar to math. The math training has helped my guitar playing tremendously and I was pretty talented before hand. The study of lifting (x) value zeros from polynomials has pushed a natural technique shift into using hammering open notes back to the root note for the respective key. An example would be the D to open E relation ship within descending riff at the end of "Extreme's Hole Hearted." chord progression. Before the math training I was perhaps playing it correctly, now I play it intelligently. You have a unique ability to explain simplicity along side complexity. Excellent work sir. Make some money at it.