Knowing when to combine terms

A common stumbling block in some student work results from multiplying factors and combining terms when the resulting function would be easier to work with if it were in factored form. For instance, it is much easier to work with

$$f(x)=\frac{(x+1)^3x(x^2+1)}{(x-1)^2(x+2)(x-2)}$$

than it would be to work with

$$f(x)= \frac{x^6+3x^5+4x^4+4x^3+3x^2+x}{x^4-2x^3-3x^2+8x-4}$$

even though they are equal. On the other hand, we do want to multiply and combine terms in

$$g(x)=\frac{(x^2-1)2x - (x^2+1)2x}{(x^2-1)^2}= \frac{-4x}{(x^2-1)^2}$$

particularly if we are going to want to find when $g(x) > 0$.

In many situations simplifications are done by either adding 0 or multiplying by 1, both suitably disguised.