1. Any questions on the Section 6.2 homework?

2. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note- taking materials.

3. Section 6.3 and 6.4 Complex Fractions and Dividing Polynomials

4. Key Concepts from Section 6.3: Complex rational expressions (complex fractions) are rational expressions whose numerator, denominator, or both contain one or more rational expressions.

5. Example of a complex fraction: 10 3x 5 6x Solution: view as a division problem: 10 ÷ 5 = 10 · 6x = 4 3x 6x 3x 5

6. Problem from today’s homework:

7. Note: This one requires that you first combine the added or subtracted pairs of rational expressions into single rational expressions with a common denominator.

8. Section 6.4 Dividing a polynomial by a monomial Divide each term of the polynomial separately by the monomial. Example

9. Problem from today’s homework:

10. Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers. This process is reviewed in detail on the next slide, but first, we’ll work these two simpler examples on the whiteboard: 1). 225 ÷ 9 2). 231 ÷ 9 Question: How can you check your answers on long division problems?

11. Divide 43 into 72. Multiply 1 times 43. Subtract 43 from 72. Bring down 5. Divide 43 into 295. Multiply 6 times 43. Subtract 258 from 295. Bring down 6. Divide 43 into 376. Multiply 8 times 43. Subtract 344 from 376. Nothing to bring down. We then write our result as 168 Example: Long Division with integers 32 43

12. As you can see from the previous example, there is a pattern in the long division technique. Divide Multiply Subtract Bring down Then repeat these steps until you can’t bring down or divide any longer. We will incorporate this same repeated technique with dividing polynomials.

13. Now you try it (And don’t forget to check your answer!) Divide 3471 by 6 using long division. Then check your answer. Do this in your notebook now, and make sure you ask if you have questions about any step. This will be crucial to your understanding of long division of polynomials.

14. 35  x Divide 7x into 28x 2. Multiply 4x times 7x+3. Subtract 28x 2 + 12x from 28x 2 – 23x. Bring down -15. Divide 7x into –35x. Multiply -5 times 7x+3. Subtract –35x–15 from –35x–15. Nothing to bring down. 15  So our answer is 4x – 5. Example with polynomials: Check: Multiply (7x + 3)(4x – 5) and see if you get 28x 2 – 23x - 15.

15. Now you try it (And don’t forget to check your answer!) Divide 6x 2 – x – 2 by 3x – 2 using long division. Then check your answer. Do this in your notebook now, and make sure to ask if you have questions about any step. ANSWER: 2x + 1

16. 86472 2  xxx x2 x x 14 4 2  20  x 10  7020  x 78 Divide 2x into 4x 2. Multiply 2x times 2x+7. Subtract 4x 2 + 14x from 4x 2 – 6x. Bring down 8. Divide 2x into –20x. Multiply -10 times 2x+7. Subtract –20x–70 from –20x+8. Nothing to bring down. 8   )72( 78  x x2 10  We write our final answer as Example

17. 86472 2  xxx x2 x x 14 4 2  20  x 10  7020  x 78 8  How do we check this answer? Final answer: 2x – 10 + 78. 2x - 7 How to check: Calculate (2x + 7)(2x – 10) + 78. If it comes out to 4x 2 – 6x + 8, then the answer is correct.

18. Now you try it (And don’t forget to check your answer!) Divide 15x 2 + 19x – 2 by 3x + 5 using long division. Then check your answer. Do this in your notebook now, and make sure you ask if you have questions about any step. Answer: 5x – 2 + 8. 3x + 5

19. Problem from today’s homework:

21. Something to be aware of on Problem #14 in today’s homework: The online “Help Me Solve This” instructions for this problem demonstrate a method called “Synthetic Division”. (We are not covering this method in class, because it is only useful in the cases where the divisor is of the form x – c.) You will get the same answer using the long division method, so this is what we expect you to use to solve this problem. (If you are already familiar with synthetic division, you’re welcome to use it in those cases where it is applicable, but we aren’t expecting you to learn it on your own.)

22. Reminder: This homework assignment on section 6.4 and 6.3 is due at the start of next class period.

23. You may now OPEN your LAPTOPS and begin working on the homework assignment.