If the numerator and denominator of a rational expression are multiplied or divided by the same factor, the result is an equivalent expression. That is, for all factors a, b, k where b ≠ 0 and k ≠ 0:
Example, divide the numerator and denominator of the expression on the left of the equality symbol by (x + 3) to get the expression on the right:a × k a —————— = ——— b × k b
Conversely, multiply the numerator and denominator of the expression on the right by (x + 3) to get the expression on the left.(x + 2)(x + 3) x + 2 ———————————————— = ——————— (x + 1)(x + 3) x + 1
A rational expression is improper if the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial. Polynomial long division can be used to write an improper rational expression as the sum of a polynomial and a proper rational expression.
Example:
x2 + 3x + 4 2
————————————— = x + 2 + ———————
x + 1 x + 1
Reduction of a rational expression to its simplest form often requires finding the greatest common divisor (GCD), sometimes known as the greatest common factor or highest common factor of two polynomials. The greatest common divisor is useful for reducing fractions to the lowest terms. For instance, in this example the GCD is 7:
Greatest common divisors can be found by determining the prime factorizations of the polynomials and comparing factors. For example, to compute the GCD of these two expressions,21 3 × 7 3 ———— = —————— = ——— 28 4 × 7 4
First, determine their prime factorizations:q(x) = x2 + 4x + 4 p(x) = x2 + 5x + 6
The largest common factor of the two expressions is (x + 2) so this is the GCD.q(x) = (x + 2)(x + 2) p(x) = (x + 2)(x + 3)
Addition and subtraction of rational expressions often require finding the least common multiple of two or more polynomials. A common multiple is a number that is a multiple of two or more numbers. The common multiples of 4 and 5 are 20, 40, 60, and so on. The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. The least common multiple of 4 and 5 is 20. This idea can be applied to polynomials. All of the factors of the each of the polynomials must be present in the factors of the LCM.
For example, the least common multiple of (x + 1) and (x + 2) is
Another example; the LCM of the factored polynomials:(x + 1)(x + 2) = x2 + 3x + 2
Is this product of factors:q(x) = (x + 2)2(x + 3) p(x) = (x + 2)(x + 3)(x + 5)
(x + 2)2(x + 3)(x + 5)