Section 6.3 Multiplying & Dividing Rational Expressions

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1 Section 6.3 Multiplying & Dividing Rational Expressions MULTIPLYING FRACTIONS In arithmetic, we can multiply fractions by multiplying the numerators separately from the denominators. For example, multiply : Multiply the numerators separately from the denominators: and we get: In algebra, we can multiply fractions by multiplying the numerators separately from the denominators. Example 1: Multiply. Simplify completely. 3x 2 4y 7x 5y b) 3x x 5x 1 Procedure: Multiply the numerators separately from the denominators to form one fraction: 3x 2 4y 7x 5y b) 3x x 5x 1 3x 2 7x 4y 5y (3x + 2) x 4 (5x 1) 21x 3 20y 2 3x 2 + 2x 20x 4 We ll often try to simplify the result of multiplication by canceling common factors, as outlined in Section 6.1. In both examples above, the resulting fractions cannot be simplified. You can see, in the second step of part (b), its factored form, and there are no common factors that can be canceled. Multiplying and Dividing Rational Expressions page 6.3-1

2 In arithmetic, when multiplying fractions, we can cross cancel common factors before actually multiplying the fractions together. For example, multiply : First factor all of the numerators and denominators: Common factors, top to bottom, can be rewritten as 1 s. The 2 s can cancel and be rewritten as 1 s: and the 3 s can cancel and be rewritten as 1 s: (The 5 s can not be rewritten as 1 s because they are both in the numerator.) We now have two fractions that can be multiplied together: and we get: In algebra, when multiplying fractions, we can cross cancel common factors before actually multiplying the fractions together. Example 2: Multiply 2x 6 x 2 + 5x x2 2x x 2 9 Procedure: First factor each numerator and denominator, then look for common factors to cross cancel. These common factors must be in the numerator and denominator. 2x 6 x 2 + 5x x2 2x x 2 9 2(x 3) x(x + 5) x(x 2) (x 3)(x + 3) The common factors of x can be rewritten as 1 s: 2(x 3) 1 (x + 5) 1 (x 2) (x 3)(x + 3) The common factors of (x 3) can be rewritten as 1 s: We now have two fractions that can be multiplied together: 2 1 1(x + 5) 2 (x + 5) 1(x 2) 1 (x + 3) (x 2) (x + 3) and we get: 2(x 2) (x + 5)(x + 3) Multiplying and Dividing Rational Expressions page 6.3-2

3 2x 4 In Example 2, we could also write the answer multiplied out as x 2 + 8x The important thing to note is that there are no more common factors that can be canceled. The final answer may be written in either factored form or polynomial form (multiplied out). The steps shown in Example 2 indicate replacing the common factors with 1 s. This might work well in the explanation of things here in the textbook but in practice, you are more likely to use the method of crosscanceling by actually crossing out the common factors, as shown here. (Notice, still, that a 1 is written above or below the canceled factor.) Exercise 1 Multiply. Factor and simplify wherever possible. 5a 2 2b 3 3a 4b 4 b) 3a 2 4b 3 14b 6a 2 c) y y 2y 3 d) 4x 2 3x 6x 10x 5 e) 3x 3 x 2 1 x2 + x 6x 9 f) 3x 2 x x 2 + 3x 10 x 2 4 7x x Multiplying and Dividing Rational Expressions page 6.3-3

4 DIVIDING FRACTIONS In arithmetic, when dividing fractions, we invert the second fraction and then multiply the fractions together. For example, divide : Invert the second fraction (write its reciprocal) and multiply : Multiply the numerators separately from the denominators: and we get: Another example, divide : Invert the second fraction (write its reciprocal) and multiply : Multiply the numerators separately from the denominators: We can replace the common factors (the 2 s and the 3 s) with 1 s: (Notice that we replaced only one of the 3 s in the second fraction; there was only one 3 to reduce from the first fraction.) 5 Multiply within the individual fractions: and we get: Multiplying and Dividing Rational Expressions page 6.3-4

5 In algebra, when dividing fractions, we invert the second fraction and then multiply the fractions together. Example 3: Divide 5x + 10 x 2 5x + 6 x 2 4 2x 2 x 15 Procedure: First: rewrite the entire problem as a multiplication by inverting the second fraction. CAUTION: DO NOT FACTOR FIRST, or even at the same time as inverting the fraction. Doing so could lead to errors. 5x + 10 x 2 5x + 6 2x2 x 15 x 2 4 Second, factor each numerator and denominator. This step requires a good understanding of how to factor polynomials. The factoring steps are not shown here. You should, however, factor each on a separate piece of paper to verify that what is shown below is accurate. 5(x + 2) (x 3)(x 2) (2x + 5)(x 3) (x 2)(x + 2) Next, replace common factors with 1 s (x 2) (2x + 5) 1 (x 2) 1 (This could also be done using cross-canceling.) Notice that the factors (x 2) are both in the denominator and cannot reduce. We now get: 5 (x 2) (2x + 5) (x 2) and multiplying, we get: 5(2x + 5) (x 2)(x 2) Since the denominator has two factors of (x 2) this answer could be written one of two ways: either as 5(2x + 5) (x 2) 2 or multiplied out as 10x + 25 x 2 4x + 4. Multiplying and Dividing Rational Expressions page 6.3-5

6 Exercise 2 Divide. Factor and simplify wherever possible. 8y 2 5x 3 4x 15y 4 b) 5a 2 2b 3 3a 4b 4 c) x 2 5x 3x 2x2 + 8x 6x d) x 3 x 2 9 x 4 x + 3 e) x 2 4 3x 8 (x + 2) f) x 2 2x 15 x 2 25 x 2 9 2x + 10 Multiplying and Dividing Rational Expressions page 6.3-6

7 We cannot forget what was learned in Section 6.1. Remember that, if two binomials one in the numerator and the other in the denominator are complete opposites of one another, they may reduce to - 1 1, or just - 1. Such is the case for 2 x x 2 shown in Example 4, below. Example 4: Divide 2 x x 2 5x x 2 4 3x 15 Procedure: First: rewrite the entire problem as a multiplication by inverting the second fraction. CAUTION: DO NOT FACTOR FIRST, or even at the same time as inverting the fraction. Doing so could lead to errors. 2 x x 2 5x 3x 15 x 2 4 Second, factor each numerator and denominator. Also, write each factor in descending order. (- x + 2) x(x 5) 3(x 5) (x 2)(x + 2) Note that (- x + 2) in the numerator and (x 2) in the denominator are complete opposites, so they can cancel to -1 1 ; of course, the factors of (x 5) also cancel: (- 1) x (1) 3 (1) (1)(x + 2) - 3 x(x + 2) Exercise 3 Apply the indicated operation. Simplify wherever possible. 3x 1 x x 1 x + 3 b) 2x 8 x x x Multiplying and Dividing Rational Expressions page 6.3-7

8 c) 3x 6 2x + 1 8x x 5x 2 d) 6 + 2x x 2 1 x x Answers to each Exercise Section 6.3 Exercise 1: 15a 3 8b 7 b) 7 4b 2 c) 3y 2 + 6y 10y 15 d) 4 5 e) x (2x 3) f) 3x 1 7x + 35 Exercise 2: 6y 6 x 4 b) 10ab 3 c) (x 5) (x + 4) d) Exercise 3: 1 (x 4) e) (x 2) (3x 8) f) 2 (x 3) - 3x (x + 3) b) - 2x (x + 5) c) x d) - 8 (x + 1) Multiplying and Dividing Rational Expressions page 6.3-8

9 Section 6.3 Focus Exercises 1. Apply the indicated operation. Simplify wherever possible. - 12p 3 5m 2 10m 4p 3 b) 8a 9b 4 4a 15b 2 c) x 2 3x 5x x d) 3x 6 4x 8x 2 x + 2 e) w 2 4 2w 8 2w + 4 w 4 f) x x x x 2 5x g) y 2 y 6 y y + 2 h) x 2 + 6x + 8 x 2 16 x + 2 2x 8 Multiplying and Dividing Rational Expressions page 6.3-9

10 2. Apply the indicated operation. Simplify wherever possible. 5x x 2 3x x2 + x 2 4x 2 b) x 2 9x + 18 x 2 11x x2 20x x 2 9 c) 2x 4 x x 4 x + 2 d) x 2 5x 2x x2 x 2 1 e) 5x x 3x x 2 f) 49 x 2 x 2 + 7x x 4 + x Multiplying and Dividing Rational Expressions page