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Multiply polynomials.

Objectives. Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers. To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents. Check It Out! Example 1. Find each product.

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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. To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

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

  6. x2 –4x1 x2 5x –2 Check It Out! Example 2b Find the product. (x2 – 4x + 1)(x2 + 5x – 2) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product.

  7. 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.

  8. Check It Out! Example 3

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

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

  11. Notice the coefficients of the variables in the final product of (a+ b)3. these coefficients are the numbers from the third row of Pascal's triangle. Each row of Pascal’s triangle gives the coefficients of the corresponding binomial expansion. The pattern in the table can be extended to apply to the expansion of any binomial of the form (a + b)n, where n is a whole number.

  12. This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.

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

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

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