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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra 2.5: Rational Expressions and Equations HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Objectives o Simplifying rational expressions. o Combining rational expressions. o Simplifying complex rational expressions. o Solving rational equations. o Interlude: work-rate problems. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Simplifying Rational Expressions A rational expression is an expression that can be written as a ratio of two polynomials P . Of course, Q such a fraction is undefined for many value(s) of the variable(s) for which Q 0 . A given rational expression is simplified or reduced when P and Q contain no common factors (other than 1 and 1). x2 x2 x 1 , Ex: x 1 x HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Simplifying Rational Expressions o To simplify rational expressions, we factor the polynomials in the numerator and denominator completely, and cancel any common factors. o However, the simplified rational expression may be defined for values of the variable(s) that the original expression is not, and the two versions are equal only where they are both defined. That is, if A , B and C represent algebraic expressions, AC A only where B 0 and C 0. BC B HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 1: Simplifying Rational Expressions Simplify the rational expression, and indicate values of the variable that must be excluded. 3 x 27 x 2 3x Step 1: Factor both polynomials. x 3 x 2 3 x 9 Step 2: Cancel common x x 3 factors. x 2 3x 9 , x 0,3 x Note: The final expression is defined only where both the simplified and original expressions are defined. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 2: Simplifying Rational Expressions Simplify the rational expression, and indicate values of the variable that must be excluded. x 2 2 x 15 5 x x 3 x 5 x 5 x3 1 x 3, x 5 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Simplifying Rational Expressions Caution! Remember that only common factors can be cancelled! A very common error is to think is to think that common terms from the numerator and denominator x4 4 can be cancelled. For instance, the statement 2 x x x4 is incorrect. is already simplified as far as 2 x possible. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Combining Rational Expressions o To add or subtract two rational expressions, a common denominator must first be found. o To multiply two rational expressions, the two numerators are multiplied and the two denominators are multiplied. o To divide one rational expression by another, the first is multiplied by the reciprocal of the second. o No matter which operation is being considered, it is generally best to factor all the numerators and denominators before combining rational expressions. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 3: Combining Rational Expressions Subtract the rational expression. 5x 2 2x x2 5x 6 x2 4 5x 2 2x x 2 x 3 x 2 x 2 x 2 x 3 5x 2 2x x 2 x 2 x 3 x 3 x 2 x 2 5x2 8x 4 2 x2 6 x x 2 x 2 x 3 x 2 x 2 x 3 3x 2 14 x 4 , x 2 x 2 x 3 x 2, 2, 3 Step 1: Factor both denominators. Step 2: Multiply to obtain the least common denominator (LCD) and solve. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: Combining Rational Expressions Add and subtract the rational expressions. x 3 x 2 x 6 2 x 2 x 10 2 x 4 x x 12 x 2 16 x 3 x 3 x 2 2 x 2 x 10 x 4 x 4 x 3 x 4 x 4 x 4 x 3 x 4 x 2 2 x2 x 10 x 4 x 4 x 4 x 4 x 4 x 4 2 x 2 x 20 2 x 2 x 10 2 x 16 x 2 16 10 2 , x 4, 3,4 x 16 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 5: Combining Rational Expressions Multiply the rational expression. x2 2 x 3 x2 2 x2 x 3x 4 x 3 x 1 x 2 x 2 x 1 x 4 x 3 x 1 x 2 x 2 x 1 x 4 x 3 x 2 , x 4, 2,1 x 2 x 4 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 6: Combining Rational Expressions Divide the rational expression. x 2 x 20 x 2 10 x 25 2x 6 x3 x 4 x 5 2x 3 6 x3 x 5 2 6 x x 4 x 5 3 2 x x 5 3x 2 x 4 x 5 2 , x 0,5 2 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Simplifying Complex Rational Expressions o A complex rational expression is a fraction in which the numerator or denominator (or both) contains at least one rational expression. For example, x 1 y 1 x or 1 x5 y o Complex rational expressions can always be rewritten as simple rational expressions. o One way to do this is to multiply the numerator and denominator by the least common denominator (LCD) of all the fractions that make up the complex rational expression. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 7: Complex Rational Expressions Simplify the complex rational expression. Step 1: Multiply the numerator and the denominator by the LCD x z x . Step 2: Cancel the common factor of z. 1 1 xz x z 1 1 x z x x z x z x z x 1 x x z z x z x z z x z x 1 , x 0, z 0, x z 0 xx z HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 8: Complex Rational Expressions Simplify the complex rational expression. 1 1 1 1 x y x y x2 y2 2 2 2 2 1 1 x y x y x2 y 2 xy 2 x 2 y 2 y x2 xy y x y x y x xy , x 0, y 0, y x 0, y x 0 yx HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Rational Equations o A rational equation is an equation that contains at least one rational expression, while any non-rational expressions are polynomials. o To solve these, we multiply each term in the equation by the LCD of all the rational expressions. This converts rational expressions into polynomials, which we already know how to solve. o However, values for which rational expressions in a rational equation are not defined must be excluded from the solution set. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 9: Solving Rational Equations Solve the rational equation. x3 6 x 2 2 x 8 2 x 4 x 12 x2 Step 1: Factor the numerators and denominators, and cancel common factors. x2 x 6 Step 2: Multiply both sides by the LCD. 2 x 8 x 2 x 6 x 2 x2 2 x 8 x 2 x 2 x 2 x 2 x2 2 x 8 x2 2 x 8 0 Step 3: Solve by factoring. Note: 2 cannot be a solution. x 2 x 4 0 x 2, 4 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 10: Solving Rational Equations 2 2 x Solve the rational equation. 1 0 4x 1 2 x2 1 0 4 x 1 4x 1 2x 4x 1 0 2 x 4 42 4 2 1 2 2 2 2 x 2 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Interlude: Work-Rate Problems o In a work-rate problem, two or more “workers” are acting in unison to complete a task. o The goal in a work-rate problem is usually to determine how fast the task at hand can be completed, either by the workers together or by one worker alone. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Interlude: Work-Rate Problems There are two keys to solving a work-rate problem: 1. The rate of work is the reciprocal of the time needed to complete the task. If a given job can be done by a worker in x units of time, the worker works at a rate of 1 jobs per unit of time. x 2. Rates of work are “additive”. This means that two workers working together on the same task have a combined rate of work that is the sum of their individual rates. Rate 1 Rate 2 Rate Together 1 1 1 Time 1 Time 2 Time Together HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 11: Work-Rate Problem One hose can fill a swimming pool in 10 hours. The owner buys a second hose that can fill the pool in half the time of the first one. If both hoses are used together, how long does it take to fill the pool? 1 The work rate of the first hose is 10 1 The work rate of the second hose is 5 1 1 1 Step 1: Set up the problem. 10 5 x Step 2: Multiply both sides by the x 2 x 10 LCD 10x , and solve. 3 x 10 1 x 3 hours 3 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 12: Work-Rate Problems The pool owner from the last example fills his empty pool with the two hoses, but accidentally turns on the pump that drains the pool also. The pump rate is slower than the combined rate of the two hoses, and the pool fills anyway, but it takes 20 hours to do so. At what rate can the pump empty the pool? 1 1 1 10 x 20 3 6 x 20 x 5 x 20 x4