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Adding and Subtracting Rational Functions: Simplifying Complex Expressions

In this chapter, we explore the process of adding and subtracting rational expressions, emphasizing the importance of like denominators and finding the least common denominator (LCD) for unlike denominators. Students will learn to identify undefined values in expressions, simplify complex fractions, and perform guided practice exercises to reinforce their understanding. With examples and step-by-step processes, this chapter equips learners with essential skills for manipulating rational functions effectively.

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Adding and Subtracting Rational Functions: Simplifying Complex Expressions

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  1. Chapter 5 5-3 Adding and subtracting rational functions

  2. Objectives Add and subtract rational expressions. Simplify complex fractions.

  3. Adding and subtracting rational expressions • Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator.

  4. + x – 3 x – 2 x + 4 x + 4 Example #1 • Add or subtract. Identify any x-values for which the expression is undefined.

  5. – 3x – 4 6x + 1 x2 + 1 x2 + 1 Example#2 • Add or subtract. Identify any x-values for which the expression is undefined.

  6. – 2x2–3x –2 3x2– 5 3x – 1 3x – 1 Example#3 • Add or subtract. Identify any x-values for which the expression is undefined.

  7. Student guided practice • Do problems 2-4 in your book page 332

  8. Adding and subtracting rational expressions • To add or subtract rational expressions with unlike denominators, first find the least common denominator (LCD). The LCD is the least common multiple of the polynomials in the denominators.

  9. Finding LCM • Find the least common multiple for each pair. • a. 4x3y7 and 3x5y4 • b. x2 – 4 and x2 + 5x + 6

  10. Adding and subtracting rational expressions • To add rational expressions with unlike denominators, rewrite both expressions with the LCD. This process is similar to adding fractions.

  11. + x – 3 2x x2 + 3x – 4 x + 4 Example#4 • Add. Identify any x-values for which the expression is undefined.

  12. + x –8 x2 – 4 x + 2 Example#5 • Add. Identify any x-values for which the expression is undefined.

  13. + 3x – 2 3x 3x – 3 2x –2 Example#6 • Add. Identify any x-values for which the expression is undefined.

  14. Example#7 • Subtract . Identify any x-values for which the expression is undefined.

  15. Student guided practice • Do problems 7-10 in your book page 332

  16. Complex fraction • Some rational expressions are complex fractions. A complex fraction contains one or more fractions in its numerator, its denominator, or both. Examples of complex fractions are shown below. Recall that the bar in a fraction represents division. Therefore, you can rewrite a complex fraction as a division problem and then simplify. You can also simplify complex fractions by using the LCD of the fractions in the numerator and denominator.

  17. x– 3 x + 2 x + 5 x – 1 Example • Simplify. Assume that all expressions are defined.

  18. 3 x x– 1 + 2 x x Example • Simplify. Assume that all expressions are defined.

  19. 20 6 3x – 3 x – 1 Example • Simplify. Assume that all expressions are defined.

  20. Student guided practice • Do problems 13-15 in your book page 332

  21. Homework • Do problems even numbers from 17-28 in your book page 332.

  22. Closure • Today we learned how to add and subtract rational expressions • Next class we are going to continue with rational functions.

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