Reducing Rational Fractions to the Common Denominator

The common denominator of few rational fractions is integral rational expression, that is divided by denominator of each fraction (see note factoring polynomials).

We usually take such common denominator, that any another common denominator is divided by chosen. So, the common denominator of fractions `x/(x+2)` and `(3x-1)/(x-2)` is polynomial `(x+2)(x-2)`. We have `x/(x+2)=(x(x-2))/((x+2)(x-2))` ; `(3x-1)/(x-2)=((3x-1)(x+2))/((x+2)(x-2))` .

We can reduce some rational fractions to the common denominator when we multiply the numerator and denominator of first fraction by `x-2` and the numerator and denominator of second fraction - by `x+2`. The polynomials `x-2` and `x+2` are called additional factors.

If we need find the common denominator for some rational fractions, we should:

  1. factor the denominator of each fraction;
  2. write the common denominator, including of all multipliers, that we obtain as a result of factoring; if we have some multiplier in several decomposition, then we will take the multiplier with the greatest exponent;
  3. find additional factors for each of the fractions (for this we divide the common denominator by denominator of fraction);
  4. multiply the numerator and denominator of each fraction by additional factor and reducing fractions to the common denominator.

Example. Reduce following fractions `a/(12a^2-12b^2)`; `b/(18a^3+18a^2b)`; `(a+b)/(24a^2-24ab)` to the common denominator.

Let′s factor the denominators:

`12a^2-12b^2=12(a-b)(a+b)`;

`18a^3+18a^2b=18a^2(a+b)`;

`24a^2-24ab=24a(a-b)`.

We should include the following multipliers into common denominator: `(a-b),(a+b),a^2` and least common multiplies of `12, 18, 24,` i.e. `LCM(12,18,24)=72`. So, the common denominator equals `72a^2(a-b)(a+b)`.

Let′s find the additional factors: for first fraction it is `6a^2` , for second is `4(a-b)`, for third is `3a(a+b)`.

So,

`(a*color(red)(6a^2))/((12a^2-12b^2)*color(red)(6a^2))=(6a^3)/(72a^2(a-b)(a+b))`;

`(b*color(green)(4*(a-b)))/((18a^3+18a^2b)*color(green)(4*(a-b)))=(4b(a-b))/(72a^2(a-b)(a+b))` ;

`((a+b)*color(blue)(3a*(a+b)))/((24a^2-24ab)*color(blue)(3a*(a+b)))=(3a(a+b)^2)/(72a^2(a-b)(a+b))`.