I have been asked to simplify this one: 12x^{3} + 8x^{2} / 6x^{2} + 4x By factoring it out I have simplified it to 4x^{2} / 2x Is there anything I can do to simplify it further? I know the rules of dividing powers and hence if it was 4x^{2} / 4x I know I could simplify it to to 4x (subtract the powers). however, I am not sure how this rule works when the coefficients of x are different as in this case.
Use parentheses! Your work so far is correct. Since there are factors in common in the numerator and denominator, your result can be simplified.
As you have it written your expression reads [tex] 12x^3 + \frac{8x^2}{6x^2} + 4x [/tex] - if this is what you intended, it does not simplify to [itex] 2x [/itex] If, on the other hand, you intended to write [tex] \frac{12x^3 + 8x^2}{6x^2 + 4x} [/tex] it still does not simplify to [itex] 2x [/itex].
This is why I have pointed out to you several times the need for parentheses. Here's how your expression should be written if you aren't able to format it nicely in LaTeX. (12x^{3} + 8x^{2}) / (6x^{2} + 4x) You are obviously a motivated math student, and have sought help here at PF a number of times. Don't make us work have to work at trying to divine what you mean by having to incorrectly interpret what you have written.
And to elaborate slightly on statdad's remark, that it still doesn't simplify to 2x, the expression 2x can be evaluated for every real value of x, while your original expression cannot be evaluated at every real x.
Remember, you can cancel items which are factors in a product , but not those that are terms in sums For example, the following is correct. [tex] \frac{2x^3+6x}{2x^2+6x}= \frac{2x(2x^2+3)}{2x(2x+3)} = \frac{2x^2+3}{2x+3} [/tex] (I cancelled a factor of [itex] 2x [/itex] from top and bottom) but the next is not correct. [tex] \frac{2x^3+6x}{2x^2+6x} = \frac{2x^3}{2x^2} [/tex] (I incorrectly cancelled [itex] 6x [/itex] from top and bottom).
Typo? You are being too kind: it's a full-blown error. Thanks for pointing it out. (The rest of my post stands, I think :) )