1. LESSON 2: THE MULTIPLICATION OF POLYNOMIALS A-SSE.A.2 A-APR.C.4 Opening Exercise 5 minutes, with explanation (2 slides) (Reminder: An area model is when you do multiplication by creating a rectangle whose sides are the factors and whose area is the product of those factors)

2. EXPLANATION OF THE OPENING EXERCISE 208

3. EXAMPLE 1

4. So let’s do that multiplication!

5. DRAWING RELATIONSHIPS Answers to these on the last slide.

6. MULTIPLICATION WITHOUT A TABLE * This is very common to do in higher levels of Algebra. As long as things are able to be grouped, you can choose to group them, and then work with the group as if it were a single variable. Distribute over the remaining terms Again, we get the same answer.

7. IMPORTANT QUESTIONS What property did we repeatedly use to multiply the binomials? How is the tabular method similar to the distributive property? Does the table work even when the binomials do not represent lengths? Why/why not? (Answer these in your notebook)

8. EXERCISE 1 AND 2 DO THESE!

9. EXERCISE 1 AND 2 (ANSWERS) Not this time! Check them in class with me / your classmates!

10. EXAMPLE 2 Find the products of the following few expressions. Can you generalize what is happening? (You can use either the distributive property or the tabular method, whichever you find most comfortable).

11. GENERALIZING THE PATTERN

12. GENERALIZING EXAMPLE 2 For the rest of this, we seem to be adding every term, and the exponent of those terms keep going down by 1 every time we do an addition. All of the middle terms cancel each other out and we’re left with just the first and the last terms being multiplied:

13. DO YOU REMEMBER THIS?

14. INTRODUCTION TO EXERCISE 3-4

15. INTRODUCTION EXPLANATION Above we have the expansion of the factors for the difference of two squares and the difference of two cubes, respectively.

16. EXERCISE 3

17. EXERCISE 3 EXPLANATION At this point, hopefully it’s becoming obvious that “FOIL” or “double bubble” is no longer an effective way to multiply polynomial expressions. These multiplications are ALL just repeated use of the distributive property. EWE: Each With Each – just make sure that each term in one polynomial is multiplied with each in the other

18. EXERCISE 4

19. ANSWERS TO EXERCISE 4 Very similarly to the other few examples, all of the middle terms appear to cancel out, and we wind up with the product of the first and last terms when we have a difference of two terms multiplied by the sum of those two terms.

20. LET’S GENERALIZE! We’re going to use a table to figure out what happens when we have the general form. Remember, when we generalize, we want to keep the same structure, but replace the things that are changing with a variable separate from the ones we may be dealing with. Remember, it’s okay to group the sign with the next monomial.

21. PROBLEM SET (PAGE 1 OF 2)

22. PROBLEM SET (PAGE 2 OF 2)