Lesson_4.1-4.2.pdf

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Baruch College, CUNY*
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MATH 146
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Feb 10, 2024
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Lesson 4.1: Rational Expressions Recall, a rational number is any number that can be written as the ration of two integers- i.e., a fraction involving integers. We would like to do the same thing with polynomials. This leads to the notion of a rational expression. Definition 0.1. A rational expression is an expression that can be written in the form p ( x ) q ( x ) , q ( x ) 6 = 0 where p ( x ) and q ( x ) are polynomial functions. The most common things we do with rational expression are: (1) simplify them (2) evaluate them (3) Find the domain of the rational function (4) Add or subtract them (5) Multiply or divide them Example 0.2. Reduce the following rational expressions to lowest terms. (1) x 2 - 9 x - 3 (2) y 2 - 5 y - 6 y 2 - 1
2 Sometimes we can factor out a negative one to cancel terms. Example 0.3. Reduce the following expressions to lowest terms: (1) a - b b - a (2) x 2 - 25 5 - x (3) x 2 - 4 x 2 - 5 x +6 (4) 5 y 2 - 13 y +6 20 y 3 +8 y 2
3 We can think of rational equation as defining a function. Example 0.4. Consider the function f ( x ) = x - 4 x - 2 . (1) Find the domain of f . (2) Find f (0), f ( - 4), f (4), f ( - 2), and f (2). Example 0.5. Consider the function g ( r ) = r 2 - 9 r +14 r 2 +3 r - 28 . (1) Find the domain of g . (2) Evaluate g (2), g ( - 3), and g ( x + h ).
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