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Chapter 7 Rational Expressions and Equations

Chapter 7 Rational Expressions and Equations. 7.1 Simplifying Rational Expressions 7.2 Multiplying & Dividing Rational Expressions 7.3 Adding & Subtracting Rational Expressions with a Common Denominator 7.4 Finding the Least Common Denominator & Forming Equivalent Rational Expressions

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Chapter 7 Rational Expressions and Equations

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  1. Chapter 7 Rational Expressions and Equations • 7.1 Simplifying Rational Expressions • 7.2 Multiplying & Dividing Rational Expressions • 7.3 Adding & Subtracting Rational Expressions with a Common Denominator • 7.4 Finding the Least Common Denominator & Forming Equivalent Rational Expressions • 7.5 Adding & Subtracting Rational Expressions with Unlike Denominators • 7.6 Complex Rational Expressions • 7.7 Rational Equations • 7.8 Models Involving Rational Equations

  2. Section 7.1 Objectives 1 Evaluate a Rational Expression 2 Determine Values for which a Rational Expression Is Undefined 3 Simply Rational Expressions

  3. Arational expression is the quotient of two polynomials. That is, a rational expression is written in the form where p and q are polynomials and q  0. Example: Evaluate for x = 4. Rational Expressions To evaluate a rational expression, replace the variable with its assigned numerical value and perform the arithmetic.

  4. Evaluating Rational Expressions Example: Evaluate for x = – 1 and y = 3.

  5. Example: Find the value for which is undefined. Find the value that would make 3x – 5 equal to 0. The rational expression is undefined when Undefined Values Because a rational expression is undefined for those values of the variable(s) that make the denominator zero, we find the values for which a rational expression is undefined by setting the denominator equal to zero and solving for the variable.

  6. Simplifying Rational Expressions Simplifying Rational Expressions If p, q, and r are polynomials, then if q 0and r  0. Steps to Simplify a Rational Expression Step 1: Completely factor the numerator and denominator of the rational expression. Step 2: Divide out the common factors.

  7. Example: Simplify: Simplifying Rational Expressions Factor the numerator. Divide out the common factor.

  8. Simplifying Rational Expressions Example: Simplify: Factor x from the numerator. Factor the numerator and denominator. Divide out the common factor.

  9. Section 7.2 Objectives 1 Multiply Rational Expressions 2 Divide Rational Expressions

  10. Multiplying Rational Expressions Steps to Multiply Rational Expressions Step 1: Factor the polynomials in each numerator and denominator. Step 2: Use the fact that if are two rational expressions, then Step 3: Divide out common factors in the numerator and denominator. Leave the remaining factors in factored form.

  11. Multiplying Rational Expressions Example: Multiply: Multiply. 1 1 5 Divide out common factors. 4 9 x2

  12. Multiplying Rational Expressions Example: Multiply: Factor each numerator and denominator. Factor again whenever possible. Divide out common factors.

  13. Dividing Rational Expressions Steps to Divide Rational Expressions Step 1: Multiply the dividend by the reciprocal of the divisor. That is, Step 2: Factor each polynomial in the numerator and denominator. Step 3: Multiply. Step 4: Divide out common factors in the numerator and denominator. Leave the remaining factors in factored form.

  14. Dividing Rational Expressions Example: Divide: Rewrite the division. Multiply the numerator by the reciprocal of the denominator. 2 Divide out common factors.

  15. Dividing Rational Expressions Example: Divide: Invert the second fraction and multiply. Factor the numerator and denominator. Divide out common factors.

  16. Section 7.3 Objectives 1 Add Rational Expressions with a Common Denominator 2 Subtract Rational Expressions with a Common Denominator 3 Add or Subtract Rational Expressions with Opposite Denominators

  17. Example: Find the sum: Adding Rational Expressions Adding Rational Expressions with a Common Denominator Step 1: Use the fact that if are two rational expressions, then Step 2: Simplify the sum by dividing out like factors.

  18. Adding Rational Expressions Example: Find the sum and simplify, if possible. Add the numerators. Factor the numerator and denominator. Divide out like factors.

  19. Example: Find the difference: Subtracting Rational Expressions Subtracting Rational Expressions with a Common Denominator Step 1: Use the fact that if are two rational expressions, then Step 2: Simplify the difference by dividing out common factors.

  20. Subtracting Rational Expressions Example: Find the difference and simplify, if possible. Subtract the numerators. Simplify. Factor the numerator and denominator. Divide out like factors.

  21. Opposite Denominators Example: Find the sum: Factor out a –1 from 4 – w Re-write. Add. Factor the numerator and denominator. Divide out like factors.

  22. Section 7.4 Objectives 1 Find the Least Common Denominator of Two or More Rational Expressions 2 Write a Rational Expression That Is Equivalent to a Given Rational Expression 3 Use the LCD to Write Equivalent Rational Expressions

  23. Finding the LCD If the denominators of a sum or difference of rational expressions are not the same, the rational expressions must be written using a least common denominator. The least common denominator (LCD) of two or more rational expressions is the polynomial of least degree that is a multiple of each denominator in the expressions. After factoring the denominators, we can see that the LCD is (2)(3)(3 – w) = 6(3 – w).

  24. Finding the LCD Finding the Least Common Denominator Step 1: Factor each denominator completely. When factoring, write the factored form using powers. For example, write x2 + 4x + 4 as (x + 2)2. Step 2: If the factors are common except for their power, then list the factor with the highest power. That is, list each factor the greatest number of times that it appears in any one denominator. Then list the factors that are not common. Step 3: The LCD is the product of the factors written in Step 2.

  25. Finding the LCD Example: Find the LCD of the rational expressions . Factor each denominator. Use the factor that is repeated the greatest number of times.

  26. Finding Equivalent Expressions Steps to Form Equivalent Rational Expressions Step 1: Write each denominator in factored form. Step 2: Determine the “missing factor(s).” That is, what factor(s) does the new denominator have that is missing from the original denominator? Step 3: Multiply the original rational expression by Step 4: Find the product. Leave the denominator in factored form.

  27. We know that 8 · 6 = 48, so we form the factor of 1 = The equivalent fraction is Finding Equivalent Expressions Example: Write as an equivalent fraction with a denominator of 48. We want to change the denominator of 8 into a denominator of 48.

  28. We know that xyz· xy = x2y2z, so we form the factor of 1 = The equivalent fraction is Finding Equivalent Expressions Example: Write the rational expression with a denominator of x2y2z. We want to change the denominator of xyz to a denominator of x2y2z.

  29. The LCD = x(x + 1)(x – 2). Writing Equivalent Expressions Example: Find the LCD of the rational expressions Rewrite each expression. x2 + x = x(x + 1) x2 – x – 2 = (x – 2)(x + 1)

  30. Section 7.5 Objectives 1 Add and Subtract Rational Expressions with Unlike Denominators

  31. Adding with Unlike Denominators Steps to Add or Subtract Rational Expressions with Unlike Denominators Step 1: Find the least common denominator. Step 2: Rewrite each rational expression as an equivalent rational expression with the common denominator. Step 3: Add or subtract the rational expressions found in Step 2. Step 4: Simplify the result.

  32. Multiply the second fraction by Adding with Unlike Denominators Example: Find the sum: The LCD is xy.

  33. Multiply the first fraction by Multiply the second fraction by Adding with Unlike Denominators Example: Find the sum: The LCD is 112x2y4.

  34. Multiply the first fraction by Subtracting with Unlike Denominators Example: Find the difference: The LCD is 2(x – 3).

  35. (d – 6)(d + 6) (d – 6)2 Subtracting with Unlike Denominators Example: Find the difference: The LCD is (d – 6)2(d + 6). Continued.

  36. Subtracting with Unlike Denominators Example continued:

  37. Section 7.6 Objectives 1 Simplify a Complex Rational Expression by Simplifying the Numerator and Denominator Separately (Method I) 2 Simplify a Complex Rational Expression Using the Least Common Denominator (Method II)

  38. Tosimplifya complex rational expression means to write it in the form where p and q are polynomials that have no common factors. One of two methods can be used: Simplifying Rational Expressions Acomplex rational expression is a fraction in which the numerator and/or the denominator contains the sum or difference of two or more rational expressions. 1. Simplifying the numerator and the denominator separately, or 2. Using the least common denominator.

  39. Simplifying Rational Expressions Simplifying a Complex Rational Expression by Simplifying the Numerator and Denominator Separately (Method I) Step 1: Write the numerator of the complex rational expression as a single rational expression. Step 2: Write the denominator of the complex rational expression as a single rational expression. Step 3: Rewrite the complex rational expression using the rational expressions determined in Steps 1 and 2. Step 4: Simplify the rational expression using the techniques for dividing rational expressions from Section 7.2.

  40. Simplifying Rational Expressions Example: Simplify: Write the denominator as a single expression. Divide. Continued.

  41. Simplifying Rational Expressions Example continued: Simplify.

  42. Simplifying Rational Expressions Simplifying a Complex Rational Expression by Using the Least Common Denominator (Method II) Step 1: Find the least common denominator among all the denominators in the complex rational expression. Step 2: Multiply both the numerator and denominator of the complex rational expression by the least common denominator found in Step 1. Step 3: Simplify the rational expression.

  43. Using the LCD to Simplify Example: Simplify: The LCD of all the denominators is 8w. Multiply each term by 8w. Simplify.

  44. Section 7.7 Objectives 1 Solve Equations Containing Rational Expressions 2 Solve for a Variable in a Rational Equation

  45. Example: Solve for x. x  0 Solving Equations Arational equation is an equation that contains a rational expression. Multiply each term by the LCD 6x. Simplify. Subtract 2 from both sides. Divide both sides by 5.  Check: The solution set is {2}.

  46. Solving Equations Steps to Solve a Rational Equation Step 1: Determine the values(s) of the variable that result in an undefined rational expression in the rational equation. Step 2: Determine the least common denominator (LCD) of all the denominators. Step 3: Multiply both sides of the equation by the LCD, and simplify the expression on each side of the equation. Step 4: Solve the resulting equation. Step 5: Verify your solution using the original equation.

  47. y – 5, y  Solving Equations Example: Solve: Multiply each term by the LCD (y + 5)(3y – 2). Simplify. Distribute to remove parentheses. Add 6 to both sides. Continued.

  48. The solution set is Solving Equations Example continued: Subtract 3y from both sides. Divide both sides by 6 and simplify. Check: 

  49. Extraneous Solution An extraneous solution is a solution that is obtained through the solving process that does not satisfy the original equation.

  50. Solving Equations with No Solutions Example: Solve: a 6 Multiply each term by (a – 6). Simplify. Distribute. Add 5a to both sides. Add 10 to both sides. Continued.

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