what's wrong here?

allegansveritatem said:
I will have to think about the middle method but I have used the first and the last. Factoring is nice...but it is applicable only when the numbers are fairly small or at least fairly neat. I mean, 2 and 4, 20 and 5 etc. or 100 and 1000 etc.
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I think you may have missed a key point. If there is no constant term, you can always factor by x, which means that x = 0 is one solution.

The reason to factor when there is no constant term is so you do not divide by by some power of x and thereby miss that zero is a solution.
 
lev888 said:
Could you clarify this limitation?
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well, maybe I'm just not much good at factoring...and I am thinking about the factoring of quadratic equations not factoring in general. When I have a quadratic to factor I usually jump to the quadratic formula, especially if there is a leading coefficient greater than 1.
 
JeffM said:
I think you may have missed a key point. If there is no constant term, you can always factor by x, which means that x = 0 is one solution.

The reason to factor when there is no constant term is so you do not divide by by some power of x and thereby miss that zero is a solution.
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yes, I can see that. Thanks. I have run into that kind of situation a couple of times just today.
 
allegansveritatem said:
well, maybe I'm just not much good at factoring...and I am thinking about the factoring of quadratic equations not factoring in general. When I have a quadratic to factor I usually jump to the quadratic formula, especially if there is a leading coefficient greater than 1.
Click to expand...
There is a theorem that says every polynomial of degree n with real coefficients can be factored into a linear term with real coefficients and (n-1)/2 quadratic terms with real coefficients if n is odd and into n/2 quadratic terms with real coefficients if n is even. Furthermore, there is a theorem that says every quadratic can be factored into 2 linear terms (though not necessarily with real coefficients.) So there is no such thing as a polynomial that cannot be factored.

Wonderful, except the theorems do not tell you how to find that factoring. What I tell my students is to look for an easy factoring of a quadratic for about a minute, but, if one does not appear quickly, revert to the quadratic formula.

There are formulas for cubics and quartics, but no one memorizes them. You can try the rational or integer root theorems, but they only apply in certain cases. After that, you are probably going to be reduced to an approximation using Newton's method or some other iterative method.

The trick, however, about seeing whether a power of x can be factored out is useful as is the trick of using a substitution to lower the degree of the equation. One or both of those tricks may let you avoid all those cumbersome methods.
 
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