Simplify 5 x x 2 − x − 6 + 4 x 2 + 4 x + 4 How come the answer is left as 5 x ( x + 2 ) ( x − 3 ) + 4 ( x + 2 ) 2 . Why don't we go any further?

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Simplify             5      x                      x        2            −      x      −      6        +      4                  x        2            +      4      x      +      4      How come the answer is left as             5      x              (      x      +      2      )      (      x      −      3      )        +      4          (      x      +      2              )        2            . Why don&#039;t we go any further?

aqueritztv · Accepted Answer

Simplifying something is to some extent not a well-defined problem, because there is a valid (and not canonically answerable) question of what actually constitutes a simpler expression for something. For instance, in the problem at hand, you can actually write it as a single fraction by obtaining a common denominator as follows:            5      x              (      x      +      2      )      (      x      −      3      )        +      4          (      x      +      2              )        2              =            5      x      (      x      +      2      )              (      x      +      2              )        2            (      x      −      3      )        +            4      (      x      −      3      )              (      x      +      2              )        2            (      x      −      3      )        =            5      x      (      x      +      2      )      +      4      (      x      −      3      )              (      x      +      2              )        2            (      x      −      3      )      One can simplify the numerator some, obtaining:            5              x        2            +      10      x      +      4      x      −      12              (      x      +      2              )        2            (      x      −      3      )        =            5              x        2            +      14      x      −      12              (      x      +      2              )        2            (      x      −      3      )      However the numerator does not factor over the rationals (      Q  ), hence it cannot be &#039;simplified&#039; any further. Now for the interesting question: Is this form really simpler than the one from your answer book? That is a really matter of opinion.Edit: To make this point through example, consider   f  (  x  ) equals the expression given in the problem, and suppose you were asked to graph   f. In that case, the form I derived is easier to use because of the theorems pertaining to the graph of a rational function given in (some?) College Algebra courses. However if you were instead asked to find an antiderivative of   f (as in Calculus), then it would be better to leave it in the form given in the answer book because that expression is much more readily integrated via   u-subs and/or partial fractions.

Simplify 5 x </mrow> x 2 </msup

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