When is the sum or difference of rational expressions simplified?

[MathMathter]

If you have a sum or difference of rational expressions, for example (8x-5)/(4x^2-5x) - (2x+2)/(x^2+2x), how do you know when it's simplified? Is there objective criteria to determine whether it is? Here, finding the LCD and combining the numerators gives us 13x^2/(4x^2-5x)(x^2+2x). That seems simpler, given that many of the terms in the combined numerators canceled, leaving only 13x^2. But what if, say, the coefficients in the numerator were different and this process just led to having a cubic polynomial in the numerator? Would that have been more simplified than what we originally had?...and why?

Thanks!

 

[lookagain]

If you have a sum or difference of rational expressions, for example (8x-5)/(4x^2-5x) - (2x+2)/(x^2+2x),

how do you know when it's simplified? Is there objective criteria to determine whether it is? H

ere, finding the LCD and combining the numerators gives us 13x^2/((4x^2-5x)(x^2+2x)). \(\displaystyle \ \ \ \) <----- You must have outer grouping symbols around the denominator.

That seems simpler, given that many of the terms in the combined numerators canceled,

leaving only 13x^2. But what if, say, the coefficients in the numerator were different and this process just led to h

aving a cubic polynomial in the numerator? Would that have been more simplified than what we originally had?...and why?

Thanks!

Factor out an x from each binomial:

\(\displaystyle \dfrac{13x^2}{(4x^2 - 5x)(x^2 + 2x)} \ = \)

\(\displaystyle \dfrac{13x^2}{[x(4x - 5)][x(x + 2)]} \ =\)


Now continue . . .

 

[MathMathter]

Right, the x's cancel. But what I'm getting at is, at the outset, how do you know if you need to find an LCD and combine the numerators over that denominator. For example, is 5/(x-3) + 2/(x+7) simplified already. Or does it need to be (7x+29)/(x-3)(x+7)?

 

[MathMathter]

Thanks for the detailed response, Romsek. That makes sense that either form would work, as I haven't found anything definitive about which is considered simplified on the internet.

Denis, right, I got sloppy and omitted the outer parentheses, which are important!