Quadratics

7.4 Factoring

Educators might be surprised that I don’t teach how to factor quadratics, that is, how to recognize:

 

\(x^{2}-5x+6\) as \((x-2)(x-3)\)

and

\(6x^{2}-7x-5\) as \((2x+1)(3x-5)\),

for example.

 

I omit this for three reasons:

 

1. Factoring yields expressions that are associated with pictures of rectangles. But quadratics are ultimately best understood through the SYMMETRY OF SQUARES. Making use of symmetry is a powerful technique in mathematics.

 

Notice, for instance, we can easily solve a symmetrical square problem: \(a^2=9\). We cannot solve, however, rectangle problems such as \(a\times b = 9\) without extra information. (There are infinitely many solutions to \(ab = 9\).)

 

Let’s not muddy the story of SYMMETRY by bringing in an additional unsymmetrical technique.

 

2. Factoring is used in the quadratics chapter of algebra books to answer questions that have been designed to be solved by factoring. This is a false construct! It gives the impression to students most quadratics can be solved by factoring, which is far from the case! Would you guess factors of the form \(1\pm \sqrt{2}\) when examining \(x^{2}-2x-3\)? Most quadratics do not factor at all nicely!

 

The box method will never let you down. Factoring, in practically all meaningful cases, will!

 

3. Factoring belongs in a different section in the study of algebra – the general study of polynomials. Why bring it in now?

The usual answer is: “Because they will need to know it later.” Hmm. If that is the case then do it later when the context for factoring exists!  Not now.

 

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