I want to set work for my 17 year old calculus students and have access to Mathematica 7. I aim to create a worksheet with rational expressions which they will then decompose into partial fractions. The structure I want to use will rely on some partial fractions appearing on more than one question (i.e. some overlap)

So I want to speed the process of adding rational fractions...

1/(2x+7)- 3/(2x-7)


which I was hoping I could resolve in Mathematica into (-26 - 7 x)/(14 + 11 x + 2 x^2) without brackets in numerator or denominator.

But alas I have work to do in every case, albeit fairly trivial.

Instead, using Together[ ... ] it yields (-26 - 7 x)/((2 + x) (7 + 2 x))

ExpandAll[%] just splits it up again, though it (horribly) multiplies the brackets in the denominators into 14 + 11 x + 2 x^2.

Surely I can add two or more rational expressions into a fully expanded rational expression in decreasing powers of x?

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You can use ExpandNumerator and ExpandDenominator in conjunction with Together.
ExpandDenominator@Together[1/(2 x + 7) - 4/(x + 2)]