9.2 Multiply Polynomials I can…multiply polynomials Students will do assigned homework. Students will study vocabulary words.

Daily Homework Quiz For use after Lesson 9.1 If the expression is a polynomial, find its degree and classify it by the number of terms. Otherwise, tell why it is not a polynomial. 1. m3 + n4m2 + m–2 No; one exponent is not a whole number. ANSWER 2. – 3b3c4 – 4b2c + c8 ANSWER 8th degree trinomial

Daily Homework Quiz For use after Lesson 9.1 Find the sum or difference. 3. (3m2 – 2m + 9) + (m2 + 2m – 4) 4m2 + 5 ANSWER 4. (– 4a2 + 3a – 1) – (a2 + 2a – 6) ANSWER –5a2 + a + 5

Multiply a monomial and a polynomial EXAMPLE 1 Multiply a monomial and a polynomial 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

GUIDED PRACTICE for Examples 1 and 2 Find the product. 1. x(7x2 +4) ANSWER 7x3 + 4x 2. 3a(2a +1) ANSWER 3a2 + 3a ANSWER 4n2 + 20n 3. 4n (n + 5)

EXAMPLE 2 Multiply polynomials using a table 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)

EXAMPLE 2 Multiply polynomials using a table STEP 2 Make a table of products. 3x2 x – 4 3x 2 – 8 – 12x 2x 3x2 x – 4 3x 2 ANSWER The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8.

GUIDED PRACTICE for Examples 1 and 2 Find the product. 1. (x+ 1)(7x +4) ANSWER   2. (a +3)(2a +1) ANSWER 2a2 + 7a + 3 ANSWER 4n2 + 19n – 5 3. (4n – 1)(n + 5)

EXAMPLE 3 Multiply polynomials vertically Find the product (b2 + 6b – 7)(3b – 4). SOLUTION STEP 1 Multiply by – 4. STEP 2 Multiply by 3b. b2 + 6b – 7 b2 + 6b – 7 3b – 4 3b – 4 – 4b2 – 24b + 28 – 4b2 – 24b + 28 3b3 + 18b2 – 21b

EXAMPLE 3 Multiply polynomials vertically STEP 3 Add products. b2 + 6b – 7 3b – 4 – 4b2 – 24b + 28 3b3 + 18b2 – 21b 3b3 + 14b2 – 45b + 28

Multiply polynomials horizontally EXAMPLE 4 Multiply polynomials horizontally 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. 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.

GUIDED PRACTICE for Examples 3, 4, and 5 Find the product. (x2 + 2x +1)(x + 2) 4. x3 + 4x2 + 5x + 2 ANSWER 5. (3y2 –y + 5)(2y – 3) ANSWER 6y3 – 11y2 + 13y – 15 6. (4b2 –5b + 6)(b – 2) ANSWER 4b2 – 13b2 + 16b – 12

Multiply polynomials horizontally EXAMPLE 4 Multiply polynomials horizontally First Outer Inner Last (2x + 3)(4x + 1) = 8x2 + 2x + 12x + 3 = (2x)(4x) + (2x)(1) + (3)(4x) + (3)(1) Write products of terms. = 8x2 + 2x + 12x + 3 Multiply. = 8x2 + 14x + 3 Combine like terms.

Multiply binomials using the FOIL pattern EXAMPLE 5 Multiply binomials using the FOIL pattern Find the product (3a + 4)(a – 2). First Outer Inner Last (3a + 4)( a - 2) = 3a2 – 6a + 4a – 8 (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.

GUIDED PRACTICE for Examples 3, 4, and 5 Find the product. (x + 3)(x + 2) 4. x2 + 5x + 6 ANSWER 5. (y + 5)(2y – 3) ANSWER 2y2 + 7y - 15 6. (4b –5)(b – 2) ANSWER 4b2 – 13b + 10

Standardized Test Practice EXAMPLE 6 Standardized Test Practice The dimensions of a rectangle are x + 3 and x + 2. Which expression represents the area of the rectangle? x2 + 6 A x2 + 5x + 6 B x2 + 6x + 6 C x2 + 6x D SOLUTION Area = length width Formula for area of a rectangle = (x + 3)(x + 2) Substitute for length and width. = x2 + 2x + 3x + 6 Multiply binomials.

Standardized Test Practice EXAMPLE 6 Standardized Test Practice = x2 + 5x + 6 Combine like terms. ANSWER The correct answer is B. A D C B CHECK You can use a graph to check your answer. Use a graphing calculator to display the graphs of y1 = (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 7 Solve a multi-step problem 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. 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? •

Solve a multi-step problem EXAMPLE 7 Solve a multi-step problem SOLUTION STEP 1 Write a polynomial using the formula for the area of a rectangle. The length is 45 – x. The width is 33 – x. Area = length width 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.

EXAMPLE 7 Solve a multi-step problem STEP 2 Substitute 3 for x and evaluate. Area = 1485 – 78(3) + (3)2 = 1260 ANSWER The area of the park is 1260 square feet.

GUIDED PRACTICE for Examples 6 and 7 The dimensions of a rectangle are x + 5 and x + 9. Which expression represents the area of the rectangle? 7 x2 + 45x A x2 + 45 B x2 + 14x + 45 C x2 + 45x + 45 D C ANSWER C

GUIDED PRACTICE for Examples 6 and 7 8. 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.

GUIDED PRACTICE for Examples 6 and 7 a. Write a polynomial that represents the combined area of the garden and the walkway. 4x2 + 38x + 90 ANSWER b. Find the combined area when the width of the walkway is 4 feet. 306 ft2 ANSWER