1. 8.7 Rational Expressions with Unlike Denominators CORD Math Mrs. Spitz Fall 2006

2. Objective Add or subtract rational expressions with unlike denominators.

3. Upcoming Midchapter Quiz Thursday 8.7 Thursday 10/26 8.8 Friday 10/27 8.9 Monday 10/30 8.10 Tuesday/Wed Chapter 8 Review Wed/Thur Chapter 8 Test Friday

4. Assignment Pp. 331-332 #3-38 all

5. Multiples – LCM What is the LCM of 4, 6 and 9? (Refer to the dancer example in your book if you want to read how it pertains to dancing and choreography). 4 – 8 – 12 – 16 – 20 – 24 – 28 – 32 – 36 – 40 6 – 12 – 18 – 24 – 30 – 36 – 42 – 48 9 – 18 – 27 – 36 – 45 – 54 So, the LCM of 4, 6 and 9 is 36.

6. What else could you do? You could use prime factorization to find LCM. 1. Find the prime factorization of each number: 2. Use each prime factor the greatest number of times it appears in any of the factorizations. 2 appears two times in 4 and 3 appears two times in 9. Thus the LCM of 4, 6, and 9 is 2  2  3  3, or 36.

7. Ex. 1: Find the LCM of 12x 2 y and 15x 2 y 2. 12x 2 y = 2  2  3  x  x  y 15x 2 y 2 = 5  3  x  x  y  y LCM = 2  2  3  5  x  x  y  y = 60x 2 y 2 –Use each factor the greatest number of times it appears in either factorization.

8. Ex. 2: Find the LCM of x 2 + x - 2 and x 2 + 5x - 6. x 2 +x - 2 = (x - 1)(x + 2) X 2 + 5x - 6 = (x + 6)(x – 1) LCM = (x – 1)(x + 2)(x + 6)–Use each factor the greatest number of times it appears in either factorization.

9. Take notes on this... To add or subtract fractions with unlike denominators, first rename the fractions so the denominators are alike. Any common denominator could be used. However, the computation is usually easier if you use the least common denominator (LCD). Recall that the least common denominator is the LCM of the denominators.

10. Use the following steps to add or subtract rational expressions. 1.Find the LCD. 2.Change each rational expression into an equivalent expression with LCD as the denominator. 3.Add or subtract as with rational expressions with like denominators. 4.Simplify if necessary.

11. Ex. 3: Find 5x = 5  x 10x 2 = 2  5  x  x ← Use each factor the greater number of times it appears. The LCD for the expression above is 2  5  x  x or 10x 2. Since the denominator of the second term is already 10x 2, only the first term needs to be renamed. ← Multiply by a representation of 1 2x/2x. ← Simplify ← Add

12. Ex. 4: Find a 2 – 4 = (a + 2)(a – 2) ← Use each factor the greater number of times it appears. ← Multiply by a representation of 1. ← Simplify ← Add a and -4a to get -3a.

13. Ex. 5: Find Multiply the first fraction by -1/-1 to change (2 – x) to (x – 2). ← Change signs. ← LCD is (x – 2)(x – 2)(x + 3) ← First term needs another (x – 2) and the second terms is missing only (x + 3).

14. Ex. 5: Find ← First term needs another (x – 2) and the second terms is missing only (x + 3). ← First term needs another (x – 2) and distribute the negative sign correctly. ← Second term needs a (x + 3). ← Combine terms.

15. Ex. 5: Find ← Combine terms. ← x 2 drop out. -2x and -2x add to give you -4x and +8 and -15 give you -7 in the numerator. In the denominator, we have what we have had throughout as the LCD (x - 2) 2 (x + 3).