Try factoring this with the quadratic formula x^5+5x^4+10x^3+10x^2+5x+1.
But this is not a trinomial and, of course, quadratic formula is not applicable here. My objection was specifically for section 4.3, which does not provide any clues how to extend the described method to more complex polynomials anyway.
The issue with formula's is that they constrain the space of the problems and ones mind especially.
That I agree with - teaching only to use formulas is bad, - but how to find polynomial roots is not relevant here. The proper way to teach factorization is IMHO to show relations to roots first - which are not necessary to be found by quadratic formula, it may be the same trial and error method, - instead of introducing a special method that works only in very limited cases.
Algebra is taught poorly not because kids don't learn the steps but because it stunts people into viewing math as if you only had the formula its easier.
Err... learning some steps (without explaining where they come from) is not a proper way to tech math anyway, and I surely do not advocate that. My point was different - instead of teaching some steps to solve particular problem (factoring trinomials), the better approach is to teach the underlying theory and real methods used in modern mathematics.
That is, instead of using factorization as a method of solving polynomial equation, the proper approach is exactly the opposite, using equation roots for the factorization. Well, at least that's how they taught me to do things, and I do believe this is the correct way.
Comments
But this is not a trinomial and, of course, quadratic formula is not applicable here. My objection was specifically for section 4.3, which does not provide any clues how to extend the described method to more complex polynomials anyway.
That I agree with - teaching only to use formulas is bad, - but how to find polynomial roots is not relevant here. The proper way to teach factorization is IMHO to show relations to roots first - which are not necessary to be found by quadratic formula, it may be the same trial and error method, - instead of introducing a special method that works only in very limited cases.
Err... learning some steps (without explaining where they come from) is not a proper way to tech math anyway, and I surely do not advocate that. My point was different - instead of teaching some steps to solve particular problem (factoring trinomials), the better approach is to teach the underlying theory and real methods used in modern mathematics.
That is, instead of using factorization as a method of solving polynomial equation, the proper approach is exactly the opposite, using equation roots for the factorization. Well, at least that's how they taught me to do things, and I do believe this is the correct way.