Rational Expressions and Functions: Adding and Subtracting 6.2 Rational Expressions and Functions: Adding and Subtracting When Denominators are the Same When Denominators are Different Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

When Denominators Are the Same Addition and Subtraction with Like Denominators To add or subtract when denominators are the same, add or subtract the numerators and keep the same denominator. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Add: Adding the numerators and combining like terms Factoring and looking for common factors Removing a factor equal to 1 Simplifying Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Recall that a fraction bar is a grouping symbol Recall that a fraction bar is a grouping symbol. The next example shows that when a numerator is subtracted, care must be taken to subtract, or change the sign of, each term in that polynomial. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Perform the indicated operation. Simplify, if possible. Solution Parentheses remind us to subtract both terms. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

When Denominators Are Different In order to add rational expressions such as we must first find common denominators. As in arithmetic, our work is easier when we use the least common multiple (LCM) of the denominators. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Least Common Multiple To find the least common multiple (LCM) of two or more expressions, find the prime factorization of each expression and form a product that contains each factor the greatest number of times that it occurs in any one prime factorization. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Find the least common multiple. We factor both expressions: By multiplying the factors of x2 + 5x + 6 by x – 7, we form a product that contains both x2 +5x +6 and x2 – 4x – 21 as factors: LCM = (x + 2)(x + 3)(x – 7) x2 + 5x + 6 is a factor. There is no need to multiply this out. x2 – 4x – 21 is a factor. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Add or Subtract Rational Expressions Determine the least common denominator (LCD) by finding the least common multiple of the denominators. Rewrite each of the original rational expressions, as needed, in an equivalent form that has the LCD. Add or subtract the resulting rational expressions, as indicated. Simplify the result, if possible, and list any restrictions on the domain of functions. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Add: The LCD is (x + 3)(x + 2)(x – 7). Multiplying by 1 to get the LCD in each expression. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution (continued) We leave the denominator in factored form. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Subtract: Noting that 3y2 – 1 and 1 – 3y2 are opposites. Performing the multiplication. Subtracting. The parentheses are important. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley