Simplify each expression using addition, subtraction, multiplication, or division.

Transcription

1 A rational epression is of the form polynomial ; a fraction of polynomial epressions. polynomial To simplify a rational epression means 1. The answer is in lowest terms ; the numerator and denominator have no common factors.. Usually, the answer may be left factored. Simplify each rational epression. 60 E 1: 0 E : E : y 5 y 15 E 4: 1 4rr r r Simplify each epression using addition, subtraction, multiplication, or division E 5: E 6: E 7: E 8: Divide: 5a 1a 4 5a 0a 4 4 a 16 a a 1

2 For addition and subtraction (as noted earlier), you need to find a least common denominator (LCD) and re-write both rational epressions with that LCD before adding or subtracting. Here are a few eamples of finding an LCD. Find the least common denominator of each problem. E 9: 1 1 n n 10 n 4n 4 n 5n 4 E 10: Add or Subtract: m 5m 40 E 11: m m m 4

3 4 8 E 1: E 1: 6 9 9

4 Many mathematicians do not consider an epression left with a radical in its denominator to be simplified. Clearing a radical from a denominator is called rationalizing the denominator. Eamine the following eample. E 14: or E 15: Rationalize the denominator: 5 8 (It is easier to rationalize the denominator if the radicals are simplified first.) E 16: Rationalize the denominator. Assume the variable is positive. 18 4

5 E 17: Rationalize the denominator. 4 9 Conjugates are epressions of the form a b and a b. When rationalizing epressions with a sum or difference in the denominator (with square roots), the numerator and denominator must be multiplied by the conjugate of the denominator. Why do we do this? Because the product of conjugates is always a rational number or epression (no roots). Eamine the following. ( 11)( 11) ( 6)( 6) When multiplying conjugates with square roots, the square root is always eliminated! When rationalizing a denominator with a binomial denominator with a square root, multiply numerator and denominator by the conjugate of the denominator. See the eample 18 below. E 18: t 5 ( t 5) ( t 5) t 5 ( t 5) ( t 5) t t t In the denominator, the 'inner' and 'outer' products were eliminated. 5 t10 t 5 t 5 t 5

6 E 19: Rationalize the denominator. 4 4 E 0: Rationalize the denominator and simplify Occasionally in calculus there is a need to rationalize the numerator. E 1: Rationalize the numerator in this epression. Simplify. 4 6

7 A comple fraction or comple rational epression is a quotient in which the numerator and/or the denominator is(are) a fractional epression or epressions. To simplify a comple rational epression, combine the numerator and/or the denominator into a single quotient (one fraction). Convert to a division problem and perform the division. E : Simplify the following epression. ( h) h E : Simplify the following. 7 7 h h 7

8 a E 4: Simplify: a a E 5: Simplify this epression. 1 a1 a 8

9 9 E 6: Simplify this comple epression. 1 1 b a a b a b E 7: Simplify: 1 4 E 8: Simplify: y y y y

10 E 9: Simplify: E 0: Simplify: ( h) 5( h) ( 5 ) h Epress each as a quotient. (Eamples 10 11) 4 E 1: E : 10

11 For this net type of eample, usually the answer should be left in factored form. E : 7 These net types of eamples will occur often in calculus when finding some types of derivatives. E 4: Factor the epression below. (6 5) ()( 4)( ) ( 4) ()(6 5) (6) E 5: Factor the epression below. 4 ( 1) ( ) ( 1)( ) 11

12 E 6: Factor the epression below. ( ) ( ) ()( ) ( ) ( ) E 7: Epress as a quotient: / 1/ E 8: Simplify the epression. (6 5) ()( 4)( ) ( 4) ()(6 5) (6) 1