The idea

Adding and subtracting algebraic fractions

Why algebraic fractions need a common denominator exactly like numeric ones, how the product of the denominators always works, and the bracket-and-sign discipline of the combined numerator.

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Algebra · Algebraic fractions

Adding and subtracting algebraic fractions

Why algebraic fractions need a common denominator exactly like numeric ones, how the product of the denominators always works, and the bracket-and-sign discipline of the combined numerator.

Why it works

Recut to a common size

\frac{2}{x}

and

\frac{3}{x+1}

are different-sized pieces —

x

-ths and

(x+1)

-ths — so, exactly as with

\frac{1}{3} + \frac{1}{4}

, they must be recut to a common size before the counts can add. The product of the denominators always serves: here

x(x + 1)

\frac{2}{x} + \frac{3}{x+1} = \frac{2(x+1) + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)}

The minus hits the WHOLE second numerator

\frac{3}{x+2} - \frac{1}{x-1} = \frac{3(x-1) - (x+2)}{(x+2)(x-1)} = \frac{3x - 3 - x - 2}{(x+2)(x-1)} = \frac{2x - 5}{(x+2)(x-1)}.

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