12.3 Multiplying Polynomials by Monomials

1. 12.3 Multiplying Polynomials by Monomials You will multiply polynomials by monomials. Essential Question: How do you multiply a polynomial by a monomial?

2. 1. ( ) 5 4x + 6 ANSWER 20x + 30 ANSWER 12n2 – 12n + 48 2. ) ( a3 + 3 – 7a 8 ANSWER 8a3 n2 – 56a + 24 + 4 – n Daily Warm-Up Exercises For use with pages 699–703 For use with pages xxx–xxx Simplify. 3. A rectangle is 12 units long and units wide. Write and simplify a polynomial expression to represent the area of the rectangle.

3. Independent Practice Check Algebra Tiles Worksheet 12.3

4. When Multiplying Polynomials by a Monomial, make sure to MultiplyALL of the Termsin the Polynomialby the Monomial • a(b + c) = ab + ac • Remember to follow the rules of Exponents Distributive Property

5. Example 1 Simplify the expression. a. b. ( ) ( ) 2n 4n2 – 5 7y3 6y5 + 9y4 SOLUTION a. Distributive property ( ) ( ) ( ) 2n 4n2 – 5 2n 4n2 – 2n 5 = Multiply coefficients. 8 n n2 – 10 n = • • • Product of powers property 8n3 – 10n = Using the Distributive Property

6. Example 1 Distributive property ( ) ( ) 7y3 6y5 + 7y3 9y4 = Multiply coefficients. 42 y3 y5 63 y3 y4 = • • + • • Product of Powers property 42y8 = + 63y7 Using the Distributive Property b. ( ) 7y3 6y5 + 9y4

7. Example 2 Simplify the expression. a. b. ( ) ( ) x3 – 3x + 8 3x2 a5 4a3 – 2a2 + a SOLUTION a. Write product. ( ) x3 – 3x + 8 3x2 Distributive property ( ) ( ) ( ) = x3 3x2 – 3x 3x2 + 8 3x2 Multiply coefficients. = 3 • x3 • x2 – 9 • x • x2 + 24 • x2 Product of powers property = 3x5 – 9x3 + 24x2 Multiplying a Monomial and a Trinomial

8. Example 2 b. Write product. ( ) a5 4a3 – 2a2 + a Distributive property ( ) ( ) ( ) = a5 4a3 – a5 2a2 + a5 a Product of powers property = 4a8 – 2a7 + a6 Multiplying a Monomial and a Trinomial

9. Example 3 Multiple Choice Practice A small outdoor music festival has reserved seats in a roped-off area in front of the main stage. The rectangular roped-off area is connected to the stage as shown below. Festival organizers have 70 feet of rope. Let l represent the length (in feet) of the roped-off area, and let w represent the width (in feet) of the roped-off area. What is a polynomial expression in terms of w that represents the area of the reserved seating?

10. Example 3 Multiple Choice Practice SOLUTION STEP 1 Write the equation given the length R of rope needed in terms of l and w. From the diagram, you can see that . (Note that there is no rope along the side of the seating area that is adjacent to the stage.) Solve this equation for l after substituting 70 for R. R l = + 2w 1 35l 70 l2 – – 2w 2 35w 70w – w2 – 2w2

11. Example 3 Multiple Choice Practice 70 l = + 2w Substitute 70 for R. 70 l – 2w = Subtract 2w from each side. Formula for area of a rectangle A lw = 70 70 ( ) – Substitute for l. 2w – 2w w = 70w Distributive property ( ) – 2w w = 70w Product of powers property – 2w2 = R l Write equation. = + 2w STEP 2 Find the area in terms of w.

12. Example 3 Multiple Choice Practice The area can be represented by the polynomial 70w – 2w2. ANSWER The correct answer is D.

13. Guided Practice 1. ( ) ANSWER 2p2 + 3p p 2p + 3 2. – – ( ) 2 x3 4x2 2x 7 + – – ANSWER 8x5 4x4 14x3 + for Examples 1, 2, and 3 Simplify the expression.

14. Guided Practice 100w – 2w2 ANSWER for Examples 1, 2, and 3 3. WHAT IF?In Example 3, suppose the organizers have 100 feet of rope to rope-off the reserved seating area. Write a polynomial expression in terms of the width w that represents the area of the reserved seating.

15. How do you multiply a polynomial by a monomial? Essential Question:

16. Benchmark Skills: • Simplifying Polynomials by Combining Like Terms • Adding Polynomials • Subtracting Polynomials Quiz Retake – Lessons 12.1 to 12.2

17. Independent Practice Textbook p. 701-702