P.8. Ratioinal Expressions
A rational expression is the ratio of two polynomials, i.e. if f is a rational expression then f can be written in the form p/q, where p and q are polynomials. Like polynomials or any other type of expression, the basic arithmetic operations, namely addition, subtraction, multiplication, and division, can be performed on rational expressions. A nice property of rational expressions is that when any of these operations are performed on two rational expressions, the result is always another rational expression. Contrary to polynomials, it is generally easy to multiply or divide but difficult to add or subtract two rational expressions.
Let f1 = p1/q1 and f2 = p2/q2 be two rational expressions, where p1, p2, q1, and q2 are polynomials. Then the product of f1 and f2 is the rational expression f1f2 = p1p2 / q1q2 and the quotient of f1 and f2 is the rational expression f1 / f2 = p1q2 / q1p2.
Example 1: Compute the product and quotient of the rational expressions (2x+5) / (3x+1) and (x-6) / (3x-4).
Solution: By multiplying the numerators and denominators of each rational expression, we see that their product is (2x+5)(x-6) / (3x+1)(3x-4). Using the rule for quotients, we see that their quotient is (2x+5)(3x-4) / (3x+1)(x-6).
When multiplying or dividing two rational expressions, it is advisable to first factor the numerator and denominator of each rational expression.
Example 2: Compute the product and quotient of the rational expressions f = (x2+3x+2) / (x2-5x+4) and g = (x2+x-2) / (x2-x-2).
Solution: Factoring the numerator and denominator of both f and g, we find f = (x+1)(x+2) / (x-1)(x-4) and g = (x+2)(x-1) / (x-2)(x+1). Their product is fg = (x+1)(x+2)2(x-1) / (x-1)(x-4)(x-2)(x+1), which simplifies to (x+2)2 / (x-4)(x-2). Their quotient is f / g = (x+1)2(x+2)(x-2) / (x-1)2(x-4)(x+2), which simplifies to (x+1)2(x-2) / (x-1)2(x-4).
Adding and subtracting rational expressions can be tricky because it is necessary to first factor the denominator of each rational expression completely. One then proceeds in much the same way as one adds or subtracts fractions, by computing the least common multiple of the denominators, rewriting each fraction over this common denominator and finally adding the resulting numerators.
Example 3: Compute the sum and difference of the rational expressions f = (x2+3x+8) / (x2+6x+5) and g = (x2+x-1) / (2x2+7x+5).
Solution: We first need to factor the denominator of each expression. The denominator of f is (x+1)(x+5) and the denominator of g is (x+1)(2x+5). The least common denominator is (x+1)(x+5)(2x+5). Thus we need to multiply the numerator of f by 2x+5 and the numerator of g by x+5. We have (x2+3x+8)(2x+5) = 2x3+11x2+31x+40 and (x2+x+1)(x+5) = x3+6x2+6x+5. Thus the numerator of the sum is 3x3+17x2+37x+45 and the numerator of the difference is x3+5x2+25x+35, so we have f+g = (3x3+17x2+37x+40) / (x+1)(x+5)(2x+5) and f-g = (x3+5x2+25x+35) / (x+1)(x+5)(2x+5).