Download 2. Rational functions and partial fractions and more Exams Algebra in PDF only on Docsity! 2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. For example x + 3 x− 7 , x− 2 2x3 + x2 − x , x2 + 3x + 2 1 . The last is the same as x2 + 3x + 2 , so any polynomial is also a rational function. If the numerator and denominator have a common factor, we can simplify the fraction by dividing top and bottom by that factor. For example, x2 + 3x + 2 x2 + 2x + 1 = (x + 1)(x + 2) (x + 1)2 = x + 2 x + 1 . To multiply two rational functions, their numerators are multiplied together and their denominators are multiplied together. To divide two rational functions, turn the second one upside-down and multiply. For example,( 4(x + 7) x + 1 ) ÷ ( x2 + 5 2x + 2 ) = ( 4(x + 7) x + 1 ) × ( 2x + 2 x2 + 5 ) = 4(x + 7)(2x + 2) (x + 1)(x2 + 5) = 8(x + 7) x2 + 5 . To add or subtract two rational functions, you must write them using a common denom- inator. For example, 1 x + 1 + 2 x + 2 = x + 2 (x + 1)(x + 2) + 2(x + 1) (x + 1)(x + 2) = x + 2 + 2(x + 1) (x + 1)(x + 2) = 3x + 4 (x + 1)(x + 2) . Note. Often the common denominator is the product of the denominators, but sometimes you can take something smaller. For example, 2 x + 1 − x (x + 1)(x + 2) = 2(x + 2) (x + 1)(x + 2) − x (x + 1)(x + 2) = 2(x + 2)− x (x + 1)(x + 2) = x + 4 (x + 1)(x + 2) . 6 2.2. Proper rational functions A proper rational function is one in which the degree of the numerator is less than the degree of the denominator. Otherwise it is called improper. Any rational function can be written as the sum of a polynomial and a proper rational function. Proof. Recall that if you divide a polynomial by a divisor, then polynomial = divisor · quotient + remainder Therefore polynomial divisor = quotient + remainder divisor . For example (using any earlier polynomial division), 2x3 + 10x2 − 3x + 1 x + 3 = (2x2 + 4x− 15) + 46 x + 3 . 2.3. Partial fractions The equality 3x + 4 (x + 1)(x + 2) = 1 x + 1 + 2 x + 2 expresses a complicated rational function as a sum of simple ones, a partial fraction. This is often useful. Example 2.1. Consider 3x− 1 (x + 1)(x− 3) . We try to write it as 3x− 1 (x + 1)(x− 3) = A x + 1 + B x− 3 where A and B are constants. Multiplying both sides by (x + 1)(x− 3) gives 3x− 1 = A(x− 3) + B(x + 1) . This is an identity which is true for all x. Putting x = 3, it gives 8 = 4B, so B = 2. Putting x = −1, it gives −4 = −4A, so A = 1. With these values of A and B the identity does hold, for 3x− 1 = (x− 3) + 2(x + 1). Therefore 3x− 1 (x + 1)(x− 3) = 1 x + 1 + 2 x− 3 . 7
