algebraic fraction

In mathematics, fraction in which letters are used to represent numbers – for example, a/b, xy/z2, and 1/(x + y). Like numerical fractions, algebraic fractions may be simplified or factorized. Two equivalent algebraic fractions can be cross-multiplied; for example, if

a/b = c/d

then ad = bc

(In the same way, the two equivalent numerical fractions 2/3 and 4/6 can be cross-multiplied to give cross-products that are both 12.)

Factorization Algebraic fractions can be simplified by factorization, that is by taking out those factors that are common to both the numerator and denominator. For example, the algebraic fraction:

(x2 − 2x − 8)(x2 + 5x + 4)/(x2 − 16)(x + 1)

can be simplified as follows:

(x2 − 2x − 8) factorizes as (x − 4)(x + 2)

(x2 + 5x + 4) factorizes as (x + 4)(x + 1)

(x2 − 16) factorizes as (x − 4)(x + 4)

so the (x − 4), (x + 4), and (x + 1) from the denominator cancel out those in the numerator, leaving only (x + 2).

Addition and subtraction As with numerical fractions, to add or subtract algebraic fractions a common denominator must be found. The easiest way to do this is to multiply the denominators together. For example:

3/x − 4/x + 2 = 3(x + 2)/ x(x + 2) − 4x/x(x + 2)

which can be simplified to:

3x +6 − 4x/x(x + 2) = 6 − x/x2 + 2x