Multiplying Polynomials 6-2 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2

Opener-SAME SHEET-10/19 Multiply. 1. x(x3) x4 2. 3x2(x5) 3x7 3. 2(5x3) 5. xy(7x2) 7x3y 6. 3y2(–3y) –9y3

6-1 Hmwk Quiz Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. A. 3 – 5x2 + 4x B. 3x2 – 4 + 8x4

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.

Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2 + 3) B. fg(f4 + 2f3g – 3f2g2 + fg3) C. 3cd2(4c2d – 6cd + 14cd2) D. x2y(6y3 + y2 – 28y + 30)

FOIL 1. (3xy + 2)(4x + 2y) 2. (x + y)(x – y)

Opener-SAME SHEET-10/20 FOIL 1. (x + 3)(4x2 – 2) 2. (3xy +2)(6x + 4y)

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.

Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) Method 1 Multiply horizontally. a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)

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

Find the product. (3b – 2c)(3b2 – bc – 2c2)

Cards Binomial Side Trinomial Side

Example 2B: Multiplying Polynomials Find the product. (y2 – 7y + 5)(y2 – y – 3) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. y2 –y –3 y2 –7y 5 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. y4 –y3 –3y2 –7y3 7y2 21y 5y2 –5y –15 y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15 y4 – 8y3 + 9y2 + 16y – 15

Check It Out! Example 2b Find the product. (x2 – 4x + 1)(x2 + 5x – 2) x4 + x3 – 21x2 + 13x – 2

Example 4: Expanding a Power of a Binomial Find the product. (a + 2b)3 (a + 2b)(a + 2b)(a + 2b) (a + 2b)(a2 + 4ab + 4b2) a3 + 6a2b + 12ab2 + 8b3

Check It Out! Example 4a Find the product. (x + 4)4 (x + 4)(x + 4)(x + 4)(x + 4) (x + 4)(x + 4)(x2 + 8x + 16) (x2 + 8x + 16)(x2 + 8x + 16) x4 + 16x3 + 96x2 + 256x + 256 Combine like terms.

Check It Out! Example 4b Find the product. (2x – 1)3 8x3 – 12x2 + 6x – 1 Combine like terms.

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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.

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

Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k – 5)3 1 3 3 1 k3 – 15k2 + 75k – 125 B. (6m – 8)3 216m3 – 864m2 + 1152m – 512

Check It Out! Example 5 x3 + 6x2 + 12x + 8 Expand each expression. a. (x + 2)3 x3 + 6x2 + 12x + 8 b. (x – 4)5 x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024

3. The number of items is modeled by Lesson Quiz Find each product. 1. 5jk(k – 2j) 5jk2 – 10j2k 2. (2a3 – a + 3)(a2 + 3a – 5) 2a5 + 6a4 – 11a3 + 14a – 15 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. –0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10 4. Find the product. (y – 5)4 y4 – 20y3 + 150y2 – 500y + 625 5. Expand the expression. (3a – b)3 27a3 – 27a2b + 9ab2 – b3