1. 6.7 Multiplying a Polynomial by a Monomial CORD Math Mrs. Spitz Fall 2006

2. Objectives: After studying this lesson, you should be able to: –Multiply a polynomial by a monomial, and –Simplify expressions involving polynomials

3. Assignment 6.7 Worksheet

4. Application The world’s largest swimming pool is the Orthlieb Pool in Casablanca, Morocco. It is 30 meters longer than 6 times its width. Express the area of the swimming pool algebraically. To find the area of the swimming pool, multiply the length by the width. Let w represent the width. Then 6w + 30 represents the length.

5. What does this look like? This diagram of the swimming pool shows that the area is w(6w + 30) square meters. This diagram of the same swimming pool shows that the area is (6w 2 + 30w) square meters. A = w(6w + 30) w 6w + 306w 30 w A = 6w 2 A = 30w Since the areas are equal, w(6w + 30) = 6w 2 + 30w The application above shows how the distributive property can be used to multiply a polynomial by a monomial.

6. Ex. 1: Find 5a(3a 2 + 4) You can multiply either horizontally or vertically. A. Use the distributive property. 5a(3a 2 + 4) = 5a(3a 2 ) + 5a(4) = 15a 3 + 20a B. Multiply each term by 5a. 3a 2 + 4 (x) 5a 15a 3 + 20a

7. Ex. 2: Find 2m 2 (5m 2 – 7m + 8) Use the distributive property. 2m 2 (5m 2 – 7m + 8) = 2m 2 (5m 2 ) + 2m 2 (-7m) + 2m 2 (8) = 10m 4 - 14m 3 + 16m 2 Ex. 3: Find -3xy(2x 2 y + 3xy 2 – 7y 3 ). Use the distributive property. -3xy (2x 2 y + 3xy 2 – 7y 3 ) = -3xy(2x 2 y) + (-3xy)(3xy 2 )+(-3xy)(-7y 3 ). = - 6x 3 y 2 - 9x 2 y 3 + 21xy 4

8. Ex. 4: Find the measure of the area of the shaded region in simplest terms. = 2a(5a 2 + 3a – 2) - 8(3a 2 – 7a + 1 = 10a 3 + 6a 2 – 4a - 24a 2 + 56a – 8 = 10a 3 - 18a 2 + 52a – 8 The measure of the area of the shaded region is 10a 3 - 18a 2 + 52a – 8. 5a 2 + 3a - 2 3a 2 - 7a + 1 2a 8

9. Ex. 5: Many equation contain polynomials that must be added, subtracted, or multiplied before the equation is solved. Solve x(x – 3) + 4x – 3 = 8x + 4 + x (3 + x) x(x – 3) + 4x – 3 = 8x + 4 + x (3 + x) x 2 – 3x + 4x – 3 = 8x + 4 + 3x + x 2 Multiply x 2 + x – 3 = x 2 + 11x + 4 Combine like terms x – 3 = 11x + 4 Subtract x 2 from each side. - 3 = 10x + 4 Subtract x from each side. - 7 = 10x Subtract 4 from each side. 7 10 - = x Check this result.