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6.1 The Fundamental Property of Rational Expressions Rational Expression – has the form: where P and Q are polynomials with Q not equal to zero. Determining when a rational expression is undefined: 1.Set the denominator equal to zero. 2.Solve the resulting equation. 3.The solutions are points where the rational expression is undefined.

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6.1 The Fundamental Property of Rational Expressions Lowest terms – A rational expression P/Q is in lowest terms if the greatest common factor of the numerator and the denominator is 1. Fundamental property of rational expressions – If P/Q is a rational expression and if K represents any polynomial where K 0, then:

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6.1 The Fundamental Property of Rational Expressions Example: Find where the following rational expression is undefined: 1.Set the denominator equal to zero. 2.Solve: 3.The expression is undefined for:

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6.1 The Fundamental Property of Rational Expressions Example: Write the rational expression in lowest terms: 1.Factor: 2.By the fundamental property: 3.The expression is undefined for:

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6.2 Multiplying and Dividing Rational Expressions Multiplying Rational Expressions – product of two rational expressions is given by: Dividing Rational Expressions – quotient of two rational expressions is given by:

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6.2 Multiplying and Dividing Rational Expressions Multiplying or Dividing Rational Expressions: 1.Factor completely 2.Multiply (multiply by reciprocal for division) 3.Write in lowest terms using the fundamental property

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6.2 Multiplying and Dividing Rational Expressions Example - multiply: Factor: Cancel to get in lowest terms:

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6.2 Multiplying and Dividing Rational Expressions Example - divide: Factor: Cancel to get in lowest terms:

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6.3 Least Common Denominators Finding the least common denominator for rational expressions: 1.Factor each denominator 2.List the factors using the maximum number of times each one occurs 3.Multiply the factors from step 2 to get the least common denominator

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6.3 Least Common Denominators Find the LCD for: 1.Factor both denominators 2.The LCD is the product of the largest power of each factor:

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6.3 Least Common Denominators Rewrite the expression with the given denominator: 1.Factor both denominators: 2.Multiply top and bottom by (p – 4)

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6.4 Adding and Subtracting Rational Expressions Adding Rational Expressions: If and are rational expressions, then Subtracting Rational Expressions: If and are rational expressions, then

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6.4 Adding and Subtracting Rational Expressions Adding/Subtracting when the denominators are different rational expressions: 1.Find the LCD 2.Rewrite fractions – multiply top and bottom of each to get the LCD in the denominator 3.Add the numerators (the LCD is the denominator 4.Write in lowest terms

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6.4 Adding/Subtracting Rational Expressions Add: 1.Factor denominators to get the LCD: 2.Multiply to get a common denominator: 3.Add and simplify:

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6.5 Complex Fractions Complex Fraction – a rational expression with fractions in the numerator, denominator or both To simplify a complex fraction (method 1): 1.Write both the numerator and denominator as a single fraction 2.Change the complex fraction to a division problem 3.Perform the division by multiplying by the reciprocal

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6.5 Complex Fractions Example: 1.Write top and bottom as a single fraction 2.Change to division problem 3.Multiply by the reciprocal and simplify

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6.5 Complex Fractions To simplify a complex fraction (method 2): 1.Find the LCD of all fractions within the complex fraction 2.Multiply both the numerator and the denominator of the complex fraction by this LCD. Write your answer in lowest terms

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6.5 Complex Fractions Example: 1.Find the LCD: the denominators are 4, 8, and x so the LCD is 8x. 2.Multiply top and bottom by this LCD. 3.Simplify:

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6.6 Solving Equations Involving Rational Expressions 1.Multiply both sides of the equation by the LCD 2.Solve the resulting equation 3.Check each solution you get – reject any answer that causes a denominator to equal zero.

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6.6 Solving Equations Involving Rational Expressions Solve: 1.Factor to get LCD LCD = x(x - 1)(x + 1) 2.Multiply both sides by LCD

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6.6 Solving Equations Involving Rational Expressions Example (continued): 3.Solve the equation 4.Check solution

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6.7 Applications of Rational Expressions Distance, Rate, and time: Rate of Work - If one job can be completed in t units of time, then the rate of work is:

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6.7 Applications of Rational Expressions Example: If the same number is added to the numerator and the denominator of the fraction 2/5, the result is 2/3. What is the number? 1.Equation 2.Multiply by LCD: 3(5+x) 3.Subtract 2x and 6

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6.7 Applications of Rational Expressions Example: It takes a mail carrier 6 hr to cover her route. It takes a substitute 8 hr. How long does it take if they work together? 1.Table: 2.Equation: 3.Multiply by LCD: 24 4.Solve: RateTimePart of Job Done Regular1/6xx/6 Substitute1/8xx/8

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