Why are quadratics factored into 2 brackets? 2x2+8x+6 2x(x+4+3×1x) or 2x(x+4+3x)?

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Why are quadratics factored into 2 brackets?  2x2+8x+6  2x(x+4+3×1x) or 2x(x+4+3x)?

Appohhl · Accepted Answer

Step 1  Factoring a constant out of quadratics in order to make it monic is fairly common.   However, factoring it in the method that you have introduces a removable singularity at x=0.   Using your example  p(x)=2x2+8x+6,  p(0)=6.  However if we call  g(x)=2x(x+8+3x),  g(0) is not defined.   Clearly p(x)=g(x)∀x≠0, but we lose continuity and differentiability at x=0 for g.

KickAntitte06 · Accepted Answer

Step 1  In fact in some contexts it is fruitful to employ such rational factorizations. For example, see this question whose solution involves rewriting symmetric polynomials as polynomials in  x+1x e.g.  ax4+bx3+2ax2+bx+a  =(a(x+1x)2+b(x+1x)+c)x2  Generally contexts enjoying some sort of innate rational structure or symmetry may similarly profit from factorizations or compositions involving rational function. You will discover some pretty examples of such if you study Galois theory.

RizerMix · Accepted Answer

Here&#039;s a few reasons: 1) No one likes division. It is difficult. 2) This factoring doesn&#039;t tell you what the roots might be. 3) Either way you have 4 &#039;&#039;terms.&#039;&#039; 4) It is not as neat. 5) Note that: 2x(x+8+3×1x)=(2x)(x+8+3×1x)

Why are quadratics factored into 2 brackets? 2x^{2}+8x+6 2x(x+4+3\times\frac{1}{x}) or 2x(x+4+\frac{3}{x})?

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