Fractions can be added or subtracted if and only if they have the same denominator. Similarly, rational expressions can be added or subtracted if and only if they have the same denominator. Thus, to add or subtract two rational expressions with unlike denominators, we must rewrite them as expressions with a common denominator. The simplest common denominator is the LCM of the denominators of each rational expression.
Example:a c a×d ± b×c ——— ± ——— = ——————————— b d b×d
To add or subtract rational expressions with like denominators, add or subtract their numerators and write the result over the denominator. Then, simplify and factor the numerator, and reduce the expression to the simplest form.4 9 4y + 9x ————— + ————— = ————————— x2y xy2 x2y2
Example:a b a ± b ——— ± ——— = ——————— c c c
2x 3y 2x + 3y ——————— + ——————— = ————————— x + y x + y x + y
To multiply two rational expressions, first factor them. Then multiply their numerators and denominators, canceling out any factors that appear in both the numerator and the denominator of the result.
Examplea c a×c ——— × ——— = ————— b d b×d
x2 + 4x + 4 x + 1 (x + 2)2(x + 1)
——————————— × ——————————— = ——————————————————————
x2 + 2x + 1 x2 + 3x + 2 (x + 1)2(x + 1)(x + 2)
x + 2
= —————————
(x + 1)2
To divide rational expressions, multiply the dividend by the reciprocal of the divisor. As above, cancel out any factors that appear in both the numerator and the denominator of the result. To obtain the reciprocal, invert the expression by exchanging the denominator with the numerator:
Example:a c a d a×d ——— ÷ ——— = ——— × ——— = ————— b d b c b×c
x + 1 x + 3 (x + 1)(x + 2) x + 1 ——————— ÷ ——————— = ———————————————— = ————————— x + 2 x + 2 (x + 2)(x + 3) x + 3