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6.1 The Fundamental Property of Rational Expressions

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: Set the denominator equal to zero. Solve the resulting equation.

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6.1 The Fundamental Property of Rational Expressions

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  1. 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: • Set the denominator equal to zero. • Solve the resulting equation. • The solutions are points where the rational expression is undefined.

  2. 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:

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

  4. 6.1 The Fundamental Property of Rational Expressions • Example: Write the rational expression in lowest terms: • Factor: • By the fundamental property: • The expression is undefined for:

  5. 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:

  6. 6.2 Multiplying and Dividing Rational Expressions • Multiplying or Dividing Rational Expressions: • Factor completely • Multiply (multiply by reciprocal for division) • Write in lowest terms using the fundamental property

  7. 6.2 Multiplying and Dividing Rational Expressions • Example - multiply: • Factor: • Cancel to get in lowest terms:

  8. 6.2 Multiplying and Dividing Rational Expressions • Example - divide: • Factor: • Cancel to get in lowest terms:

  9. 6.3 Least Common Denominators • Finding the least common denominator for rational expressions: • Factor each denominator • List the factors using the maximum number of times each one occurs • Multiply the factors from step 2 to get the least common denominator

  10. 6.3 Least Common Denominators • Find the LCD for: • Factor both denominators • The LCD is the product of the largest power of each factor:

  11. 6.3 Least Common Denominators • Rewrite the expression with the given denominator: • Factor both denominators: • Multiply top and bottom by (p – 4)

  12. 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

  13. 6.4 Adding and Subtracting Rational Expressions • Adding/Subtracting when the denominators are different rational expressions: • Find the LCD • Rewrite fractions – multiply top and bottom of each to get the LCD in the denominator • Add the numerators (the LCD is the denominator • Write in lowest terms

  14. 6.4 Adding/Subtracting Rational Expressions • Add: • Factor denominators to get the LCD: • Multiply to get acommon denominator: • Add andsimplify:

  15. 6.5 Complex Fractions • Complex Fraction – a rational expression with fractions in the numerator, denominator or both • To simplify a complex fraction (method 1): • Write both the numerator and denominator as a single fraction • Change the complex fraction to a division problem • Perform the division by multiplying by the reciprocal

  16. 6.5 Complex Fractions • Example: • Write top and bottom as a single fraction • Change to division problem • Multiply by thereciprocal and simplify

  17. 6.5 Complex Fractions • To simplify a complex fraction (method 2): • Find the LCD of all fractions within the complex fraction • Multiply both the numerator and the denominator of the complex fraction by this LCD. Write your answer in lowest terms

  18. 6.5 Complex Fractions • Example: • Find the LCD: the denominators are 4, 8, and x so the LCD is 8x. • Multiply top and bottom by this LCD. • Simplify:

  19. 6.6 Solving Equations Involving Rational Expressions • Multiply both sides of the equation by the LCD • Solve the resulting equation • Check each solution you get – reject any answer that causes a denominator to equal zero.

  20. 6.6 Solving Equations Involving Rational Expressions • Solve: • Factor to get LCDLCD = x(x - 1)(x + 1) • Multiply both sides by LCD

  21. 6.6 Solving Equations Involving Rational Expressions • Example (continued): • Solve the equation • Check solution

  22. 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:

  23. 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? • Equation • Multiply by LCD: 3(5+x) • Subtract 2x and 6

  24. 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? • Table: • Equation: • Multiply by LCD: 24 • Solve:

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