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Multiplying Polynomials Warm Up Lesson

This lesson presentation introduces the concept of multiplying polynomials through examples and guided practice. Students will learn to simplify and find the product of polynomial expressions. The lesson concludes with a quiz to assess understanding.

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Multiplying Polynomials Warm Up Lesson

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  1. Multiply Polynomials Warm Up Lesson Presentation Lesson Quiz

  2. –18a + 2b ANSWER 5200 ANSWER 2.Simplifyr2s rs3. r3s4 ANSWER Warm-Up 1.Simplify –2 (9a – b). 3. The number of hardback h and paperback p books (in hundreds) sold from 1999–2005 can be modeled by h = 0.2t2 – 1.7t + 14 and p = 0.17t3 – 2.7t2 + 11.7t + 27 where t is the number of years since 1999. About how many books sold in 2003?

  3. Example 1 Find the product 2x3(x3 + 3x2 – 2x + 5). 2x3(x3 + 3x2 – 2x + 5) Write product. = 2x3(x3) + 2x3(3x2) – 2x3(2x) + 2x3(5) Distributive property = 2x6 + 6x5 – 4x4 + 10x3 Product of powers property

  4. Example 2 Find the product(x – 4)(3x + 2). SOLUTION STEP 1 Write subtraction as addition in each polynomial. (x – 4)(3x + 2) = [x + (– 4)](3x + 2)

  5. ANSWER The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8. 2 3x x 3x2 – 4 2 3x x 3x2 – 4 Example 2 STEP2 Make a table of products. 2x – 12x – 8

  6. ANSWER ANSWER 7x3 + 4x 2a2 + 7a + 3 4n2 + 19n – 5 ANSWER Guided Practice Find the product. 1. x(7x2 +4) 2. (a +3)(2a +1) 3. (4n – 1)(n + 5)

  7. STEP1 STEP2 Multiply by– 4. Multiply by3b. 3b – 4 3b– 4 Example 3 Find the product (b2 + 6b – 7)(3b – 4). SOLUTION b2 + 6b – 7 b2 + 6b – 7 –4b2 – 24b + 28 – 4b2 – 24b + 28 3b3+ 18b2– 21b

  8. STEP3 Add products. b2 + 6b – 7 3b – 4 – 4b2 – 24b + 28 3b3 + 18b2 – 21b Example 3 3b3 + 14b2 – 45b + 28

  9. Example 4 Find the product (2x2 + 5x – 1)(4x – 3). (2x2 + 5x – 1)(4x – 3) Write product. = 2x2(4x – 3) +5x(4x – 3) – 1(4x – 3) Distributive property = 8x3 – 6x2 + 20x2 – 15x – 4x + 3 Distributive property = 8x3 + 14x2 – 19x + 3 Combine like terms.

  10. Last Outer Inner First Example 4 FOIL PATTERN The letters of the word FOIL can help you to remember how to use the distributive property to multiply binomials. The letters should remind you of the words First, Outer, Inner, and Last. (2x + 3)(4x + 1) = 8x2 + 2x + 12x + 3

  11. Example 5 Find the product(3a + 4)(a – 2). (3a + 4)(a – 2) = (3a)(a) + (3a)(– 2) + (4)(a) + (4)(– 2) Write products of terms. = 3a2 + (– 6a) + 4a + (– 8) Multiply. = 3a2 – 2a – 8 Combine like terms.

  12. 4. (x2 + 2x +1)(x + 2) ANSWER x3 + 4x2 + 5x + 2 ANSWER 6y3 – 11y2 + 13y – 15 ANSWER 4b2 – 13b + 10 Guided Practice Find the product. 5. (3y2 –y + 5)(2y – 3) 6. (4b –5)(b – 2)

  13. ANSWER The correct answer is B. x2 + 5x + 6 B x2 + 6x + 6 x2 + 6x x2 + 6 A C B D A D C Area = length width Example 6 The dimensions of a rectangle are x + 3 and x + 2.Which expression represents the area of the rectangle? SOLUTION Formula for area of a rectangle = (x + 3)(x + 2) Substitute for length and width. =x2 + 2x + 3x + 6 Multiply binomials. = x2 + 5x + 6 Combine like terms.

  14. CHECK You can use a graph to check your answer. Use a graphing calculator to display the graphs ofy1=(x + 3)(x + 2) and y2=x2 + 5x + 6 in the same viewing window. Because the graphs coincide, you know that the product of x + 3 and x + 2 is x2 + 5x + 6. Example 6

  15. The dimensions of a rectangle are x + 5 and x + 9. Which expression represents the area of the rectangle? x2 + 45 x2 + 14x + 45 x2 + 45x + 45 x2 + 45x C A D B C 7 ANSWER C Guided Practice

  16. • Write a polynomial that represents the area of the skateboard park. What is the area of the park if the walkway is 3 feet wide? • Example 7 SKATEBOARDING You are designing a rectangular skateboard park on a lot that is on the corner of a city block. The park will have a walkway along two sides. The dimensions of the lot and the walkway are shown in the diagram.

  17. Area=lengthwidth Example 7 SOLUTION STEP1 Write a polynomial using the formula for the area of a rectangle. The length is 45–x. The width is 33 – x. Formula for area of a rectangle = (45 – x)(33 – x) Substitute for length and width. = 1485 – 45x – 33x + x2 Multiply binomials. = 1485 – 78x + x2 Combine like terms.

  18. ANSWER The area of the park is1260square feet. Example 7 STEP2 Substitute 3 for x and evaluate. Area= 1485 – 78(3) + (3)2 = 1260

  19. ANSWER 4x2 + 38x + 90 ANSWER 306 ft2 Guided Practice GARDEN DESIGN You are planning to build a walkway that surrounds a rectangular garden, as shown. The width of the walkway around the garden is the same on every side. a. Write a polynomial that represents the combined area of the garden and the walkway. b. Find the combined area when the width of the walkway is 4 feet.

  20. ANSWER 12x2 + x – 6 ANSWER 3x4 – 9x3 + 6x2 – 12x 3b3 – 11b2 + 7b + 5 ANSWER ANSWER 2y2 – 3y – 20 Lesson Quiz Find the product. 1. 3x(x3 – 3x2 + 2x – 4) 2. (y – 4)(2y + 5) 3. (4x + 3)(3x – 2) 4. (b2 – 2b – 1)(3b – 5)

  21. ANSWER 3x2 + 11x – 4 Lesson Quiz 5. The dimensions of a rectangle arex + 4 and 3x – 1. Write an expression to represent the area of the rectangle.

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