University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

181

Pages:

11
Author:

timeforjamvyr

§ 14.1 Simplifying Rational Expressions Rational Expressions Rational expressions can be written in the form where P and Q are both polynomials and Q ≠ 0. P Q Examples of Rational Expressions 3x 2 + 2 x − 4 4x − 5 4x + 3y 2 2 2 x − 3xy + 4 y 3x 2 4 4 Evaluating Rational Expressions To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result. Example Evaluate the following expression for y = −2. y−2 −4 4 −2 − 2 = = = − 5 + y −5 + (−2) − 7 7 5 Evaluating Rational Expressions In the previous example, what would happen if we tried to evaluate the rational expression for y = 5? y−2 3 5 − 2 = = − 5 + y −5 + 5 0 This expression is undefined! 6 Undefined Rational Expressions We have to be able to determine when a rational expression is undefined. A rational expression is undefined when the denominator is equal to zero. The numerator being equal to zero is okay (the rational expression simply equals zero). 7 Undefined Rational Expressions Find any real numbers that make the following rational expression undefined. Example 9 x3 + 4 x 15 x + 45 The expression is undefined when 15x + 45 = 0. So the expression is undefined when x = −3. 8 Simplifying Rational Expressions Simplifying a rational expression means writing it in lowest terms or simplest form. To do this, we need to use the Fundamental Principle of Rational Expressions If P, Q, and R are polynomials, and Q and R are not 0, PR P = QR Q 9 Simplifying Rational Expressions Simplifying a Rational Expression 1) Completely factor the numerator and denominator. 2) Apply the Fundamental Principle of Rational Expressions to eliminate common factors in the numerator and denominator. Warning! Only common FACTORS can be eliminated from the numerator and denominator. Make sure any expression you eliminate is a factor. 10 Simplifying Rational Expressions Example Simplify the following expression. 7 x + 35 7( x + 5) 7 = = 2 x( x + 5) x + 5x x 11 Simplifying Rational Expressions Example Simplify the following expression. 2 ( x + 4)( x − 1) x + 3x − 4 x −1 = = 2 x − x − 20 ( x − 5)( x + 4) x −5 12 Simplifying Rational Expressions Example Simplify the following expression. 7 − y − 1( y − 7) = −1 = y−7 y−7 13