1. BY: ANNA SMOAK Multiplying a Polynomial by a Monomial

2. Class Background Algebra 1 Already learned the distributive property and the laws of exponents Goal: Review both of these topics and then have students work together to generalize their knowledge to a more abstract setting

3. Given:3x  What does this mean?  Is there another way to write 3x without using the property of multiplication?  3x = x + x + x Given:3(x + 1)  What does this mean? How is this different?  Is there another way to write 3(x+1) without using the property of multiplication?  3(x + 1) = (x + 1) + (x + 1) + (x + 1)  Now simplify this expression:  3(x + 1) = (x + 1) + (x + 1) + (x + 1) = x + x + x + 1 + 1 + 1 = 3x + 3

4. Given:2(x - 3)  What does this mean?  Is there another way to write 2(x – 3) without using the property of multiplication?  2(x – 3) = (x – 3) + (x – 3)  Now simplify this expression:  2(x – 3) = (x – 3) + (x – 3) = x + x – 3 – 3 = 2x – 6 We have seen that 3(x + 1) = 3x + 3 2(x – 3) = 2x – 6 What property can we use to get from our original expression to our simplified expression? The Distributive Property

5. Given -2(x + 1), use the distributive property to simplify the expression.  -2(x + 1) = -2x- 2

6. Simplify 3 2 x 3 4  How can you write 3 2 without using exponents? What does it mean for something to be squared?  3 2 = 3 x 3  How can you write 3 4 without using exponents? What does it mean for something to be raised to the fourth power?  3 4 = 3 x 3 x 3 x 3  Now multiply 3 2 x 3 4 using our expanded notation  What property can we use to get from our original expression to our simplified expression? Product of Powers 3 2 x 3 4 = (3 x 3) x (3 x 3 x 3 x 3) = 3636

7. What do we do with our coefficients when we multiply two monomials such as (y 4 )(12y 7 )?  We multiply our coefficients What do we do with our variables?  We add their exponents So what is (y 4 )(12y 7 )?  (1 x 12)(y 4 + 7 )=12y 11 Using the product of powers how would we simplify (4ab 6 ) (7a 2 b 3 )?  (4ab 6 ) (7a 2 b -3 ) = (4 x 7)(a 1 + 2 )(b 6 +3 ) = 28a 3 b 9

8. IN PAIRS Discuss the difference between 3x (2x + 5) and 3 (2x + 5)

9. Given 3x 2 (2x 2 + 5x – 1) discuss in pairs how you think we could simplify this expression.  We must use the distributive property to distribute the 3x 2 to each term in (2x 2 + 5x – 1).  To perform this distribution we must use the rule of product of powers.  3x 2 (2x 2 + 5x – 1) = = 6x 4 + 15x 3 - 3x 2 3x 2 (2x 2 )+ 3x 2 (5x)+ 3x 2 (-1)

10. IN PAIRS Simplify - (x – 2) + x (6x – 7)  What is our first step?  - (x – 2) + x (6x – 7)  Are we finished now? We must collect like terms o Are we finished now? We must simplify o Are we finished now? We must arrange our terms o Are we finished now? How do you know? = -x + 2 + 6x 2 – 7x = -x – 7x + 2 + 6x 2 = 6x 2 – 8x + 2 = -8x + 2 + 6x 2

11. IN PAIRS Simplify: 6rs(r 2 s - 3)= 6rs (r 2 s) + 6rs (-3)= 6r 3 s 2 - 18rs Simplify:-9x (x 2 + xy - 2y 2 )= -9x (x 2 ) - 9x (xy) - 9x (- 2y 2 )= -9x 3 - 9x 2 y + 18xy 2

12. TICKET OUT OF THE DOOR When a monomial is multiplied by a binomial, will the product always, sometimes, or never be a binomial?