Rational Fraction and its Basic Property

We can write any fractional algebraic expression in the form of `P/Q` , where `P` and `Q` are rational expressions, besides `Q` obligatory contains the variables. Such fraction is called rational fraction.

Examples of rational fractions: `(x+1)/(2x-1/3)` , `((x+2)(x^2-3))/(a+2b+5c)` , `(a/b+c/d)/(a-b)` .

The basic property of rational fraction is denominated by identity `color(blue)(P/Q=(PR)/(QR))` , that is correct for `R!=0` and `Q!=0` and `R` is integral rational expression.

This means that both numerator can be multiplied or divided by the same non- zero number, monomial or polynomial.

So, `(1/3x^3-1/2x^2+1)/(1/4x^2+1/6x+1/2)=(12(1/3x^3-1/2x^2+1))/(12(1/4x^2+1/6x+1/2))=(4x^3-6x^2+12)/(3x^2+2x+6)`.

We can use main property of fraction to change signs of both numerator and denominator. If we multiply the numerator and denominator of fraction `P/Q` by `-1`, then we will obtain `P/Q=(-P)/-Q` .

Thereby, the value of fraction will not change, if we at the same time change the signs of numerator and denominator. If we change only the sign of numerator or only the sign of denominator, then the fraction will change the sign: `(-P)/Q=-P/Q` ; `P/(-Q)=-P/Q` .

It is following that `color(red)(P/Q=-(-P)/Q=-P/(-Q))` .

For example, `(3x-2)/(3x+4)=-(-(3x-2))/(3x+4)=-(2-3x)/(3x+4)` .