Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems: College Algebra

2.5: Rational Expressions and Equations




Objectives

o Simplifying rational expressions.o Combining rational expressions.o Simplifying complex rational expressions.o Solving rational equations.o Interlude: work-rate problems.




Simplifying Rational Expressions

A rational expression is an expression that can be

written as a ratio of two polynomials . Of course,

such a fraction is undefined for many value(s) of the

variable(s) for which . A given rational

expression is simplified or reduced when and

contain no common factors (other than and ).

Ex:

PQ

0Q

P Q

1 122 1,

1x x xx x





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 , and represent algebraic expressions,

.

A BC

only where 0 and 0AC A B CBC B




Example 1: Simplifying Rational Expressions

Simplify the rational expression, and indicate values of the variable that must be excluded. 3

2

273

xx x

23 3 93

x x xx x

2 3 9 , 0,3x x xx

Step 1: Factor both polynomials.

Step 2: Cancel common factors.

Note: The final expression is defined only where both the simplified and original expressions are defined.




Example 2: Simplifying Rational Expressions

Simplify the rational expression, and indicate values of the variable that must be excluded.

2 2 155

x xx

3 5

5x x

x

31

x

3, 5x x





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

can be cancelled. For instance, the statement

is incorrect. is already simplified as far as

possible.

2

4 4xx x

2

4xx




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.




Example 3: Combining Rational Expressions

Subtract the rational expression.

2 2

5 2 25 6 4x x

x x x

5 2 22 3 2 2x x

x x x x

2 35 2 22 2 3 3 2 2

x xx xx x x x x x

2 25 8 4 2 6

2 2 3 2 2 3x x x x

x x x x x x

23 14 4 , 2, 2, 3

2 2 3x x x

x x x

Step 1: Factor both denominators.

Step 2: Multiply to obtain the least common denominator (LCD) and solve.





Add and subtract the rational expressions.2 2

2 2

3 6 2 104 12 16

x x x x xx x x x

23 23 2 104 4 3 4 4

x xx x xx x x x x

24 3 4 2 2 104 4 4 4 4 4

x x x x x xx x x x x x

2 2

2 2

2 20 2 1016 16

x x x xx x

2

10 , 4, 3,416

xx





Multiply the rational expression. 2

2

2 3 22 3 4

x x xx x x

3 1 22 1 4

x x xx x x

3 1 22 1 4

x x xx x x

3 2, 4, 2,1

2 4x x

xx x





Divide the rational expression.2 2

3

20 10 252 6

x x x xx x

3

2

4 5 62 5

x x xx x

3

2

6 4 5

2 5

x x x

x x

3 2

23 4, 0,5

5x x

xx




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,

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.

1 1

orx yx

15xy




Example 7: Complex Rational Expressions

Simplify the complex rational expression. 1 1x z xz

1 1

1

x z xx z xz x z x

x x zz x z x

z

z x z x

1 , 0, 0, 0x z x z

x x z

Step 1: Multiply the numerator and the denominator by the LCD . x z x

Step 2: Cancel the common factor of . z




Example 8: Complex Rational Expressions

Simplify the complex rational expression.

1 1

2 2

x yx y

2 2

2 2

2 2

1 1

1 1x yx yx y

x y

2 2

2 2

xy x yy x

xy y xy x y x

, 0, 0, 0, 0xy x y y x y xy x




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.




Example 9: Solving Rational Equations

Solve the rational equation. 3 2

2

6 2 84 12 2

x x xx x x

2 6 2 82 6 2

x x xx x x

2 2 82 22 2

x xx xx x

2 2 8x x

2 2 8 0x x

2 4 0x x

2, 4x

Step 1: Factor the numerators and denominators, and

cancel common factors. Step 2: Multiply both sides by

the LCD.

Step 3: Solve by factoring.

Note: cannot be a solution. 2




Example 10: Solving Rational Equations

Solve the rational equation. 22 1 04 1xx

224 1 1 0

4 1xxx

22 4 1 0x x

24 4 4 2 12 2

x

2 22

x




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.




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 units of time, the worker works at a rate of jobs per unit of time.

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.

x1x

1Time

1Ti

1Ti me 1 Tome ge

er2 th

Rate 1 Rate 2 Rate Together




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?The work rate of the first hose is

110

The work rate of the second hose is 15 1 1 1

10 5 x Step 1: Set up the problem.

Step 2: Multiply both sides by the LCD , and solve. 10x

2 10x x

3 10x 133

x hours




Example 12: Work-Rate ProblemsThe 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 1 110 203

x

6 20x x 5 20x

4x

Hawkes Learning Systems: College Algebra

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