Adding and subtracting algebraic fractions

Factor denominators first and choose a common denominator containing every required factor. Multiply each numerator by the same factor used to change its denominator. A subtraction sign applies to the whole second numerator.

Example

For \(x\ne2,3\), \(\begin{aligned}\dfrac1{x-2}-\dfrac2{x-3}&=\dfrac{(x-3)-2(x-2)}{(x-2)(x-3)}\\&=\dfrac{1-x}{(x-2)(x-3)}.\end{aligned}\)

When a denominator is quadratic, use its factors: for \(x\ne-3,3\), \(\begin{aligned}\dfrac1{x^2-9}+\dfrac2{x-3}&=\dfrac{1+2(x+3)}{(x-3)(x+3)}\\&=\dfrac{2x+7}{x^2-9}.\end{aligned}\)

A repeated factor must appear enough times: the common denominator of \(\dfrac1{(x-3)^2}\) and \(\dfrac2{x-3}\) is \((x-3)^2\). Their sum is \(\dfrac{2x-5}{(x-3)^2}\), with \(x\ne3\).