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Multiply polynomials.

Learn how to multiply polynomials and expand binomial expressions with step-by-step examples and practical applications in business. Practice quizzes included.

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Multiply polynomials.

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  1. Objectives Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

  2. Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2+ 3) B. fg(f4 + 2f3g – 3f2g2 + fg3)

  3. Check It Out! Example 1 Find each product. a. 3cd2(4c2d– 6cd + 14cd2) b. x2y(6y3 + y2 – 28y + 30)

  4. To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first. Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

  5. Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2)

  6. Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2)

  7. Example 2B: Multiplying Polynomials Find the product. (y2 – 7y + 5)(y2 – y – 3)

  8. Check It Out! Example 2a Find the product. (3b – 2c)(3b2 – bc – 2c2)

  9. Check It Out! Example 2b Find the product. (x2 – 4x + 1)(x2 + 5x – 2)

  10. Example 3: Business Application A standard Burly Box is p ft by 3p ft by 4p ft. A large Burly Box has 1.5 ft added to each dimension. Write a polynomial V(p) in standard form that can be used to find the volume of a large Burly Box.

  11. Check It Out! Example 3 Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 0.02x2 + 0.2x + 3. The average cost per unit (in dollars) can be modeled by C(x) = –0.004x2 – 0.1x + 3. Write a polynomial T(x) that can be used to model the total costs.

  12. Example 4: Expanding a Power of a Binomial Find the product. (a + 2b)3

  13. Check It Out! Example 4a Find the product. (x + 4)4

  14. Check It Out! Example 4b Find the product. (2x – 1)3

  15. Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k –5)3

  16. Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. B. (6m – 8)3

  17. Check It Out! Example 5 Expand each expression. a. (x +2)3 b. (x –4)5

  18. Check It Out! Example 5 Expand the expression. c. (3x +1)4

  19. Lesson Quiz Find each product. 1. 5jk(k – 2j) 2. (2a3– a + 3)(a2+ 3a – 5) 3. The number of items is modeled by 0.3x2 + 0.1x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost. 4. Find the product. (y – 5)4 5. Expand the expression. (3a – b)3

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