Name Class Date Multiplying Polynomials Going Deeper Essential question: How do you multiply polynomials? 6-5 A monomial is a number, a variable, or the product of a number and one or more variables raised to whole number powers, such as 5,, -8y, and 3 2 y 4. A polynomial is a monomial or a sum of monomials. Each monomial in the epression is called a term. A polynomial with two terms is a binomial. You can multiply two binomials by using algebra tiles. 1 A-SSE.1.2 EXPLORE Multiplying Two Binomials Using Algebra Tiles To use algebra tiles to multiply (2 + 1)( + 3), first represent 2 + 1 vertically along the left side of an algebra tile diagram and + 3 horizontally along the top. Then use 2 -tiles, -tiles, and 1-tiles to complete the diagram, as shown below. + 3 1 1 1 2 2( + 3) = 2 + 1( + 3) = + + 1 1 1 1 1 (2 + 1)( + 3) = 2 + + The product is a trinomial, a polynomial with three terms. REFLECT 2 + + 1a. Look at the algebra tile diagram. What two terms in the original binomials combine to form the 2 -term in the trinomial? How do they combine (by multiplying, by adding, or by subtracting)? 1b. Look at the algebra tile diagram. What two terms in the original binomials combine to form the constant term in the trinomial? How do they combine (by multiplying, by adding, or by subtracting)? Chapter 6 349 Lesson 5
1c. Look at the algebra tile diagram. Show how the terms of the original binomials combine to form the -term in the trinomial. 1d. You can verify that the epressions are equivalent by substituting a value for into both epressions and simplifying to show that they are equal. Verify that the epressions are equivalent. Use = 4. 1e. Suppose you want to use algebra tiles to find the product (2 + 1)( + 2). Describe how you can modify the algebra tile diagram to find the product. 1f. Suppose you want to use algebra tiles to find the product (2 + 2)( + 3). Describe how you can modify the algebra tile diagram to find the answer. 2 A-APR.1.1 ENGAGE Multiplying Binomials Using the Distributive Property Using algebra tiles to multiply two binomials is a useful tool for understanding how the two binomials are being multiplied. However, it is not a very practical method for everyday use. Using the distributive property is. To multiply (2 + 1)( + 3) using the distributive property, you distribute the binomial + 3 to each term of 2 + 1. Then you distribute the monomial 2 to each term of + 3 as well as the monomial 1 to each term of + 3. (2 + 1)( + 3) = 2( + 3) + 1( + 3) = 2 2 + 6 + + 3 = 2 2 + 7 + 3 Notice that the product found using algebra tiles in the Eplore is the same as the product found here using the distributive property. Thus, the two methods are equivalent. To multiply (4-7)(3 + 6) using the distributive property, you should think of 4-7 as 4 + (-7) and therefore keep the negative sign with the 7. (4-7)(3 + 6) = 4(3 + 6) - 7(3 + 6) = 12 2 + 24-21 - 42 = 12 2 + 3-42 Chapter 6 350 Lesson 5
This method of using the distributive property to multiply two binomials is referred to as the FOIL method. The letters of the word FOIL stand for First, Outer, Inner, and Last and will help you remember how to use the distributive property to multiply binomials. You apply the FOIL method by multiplying each of the four pairs of terms described below and then simplifying the resulting polynomial. First refers to the first terms of each binomial. Outer refers to the two terms on the outside of the epression. Inner refers to the two terms on the inside of the epression. Last refers to the last terms of each binomial. Now multiply (7-1)(3-5) using FOIL. Again, think of 7-1 as 7 + (-1) and 3-5 as 3 + (-5). This results in a positive constant term of 5 because (-1)(-5) = 5. Outer Inner First Inner (7-1)(3-5) = 21 2-35 - 3 + 5 First Last Outer Last (7-1)(3-5) = 21 2-38 + 5 Notice that the trinomials are written with variable terms in descending order of eponents and with the constant term last. This is a standard form for writing polynomials: Starting with the variable term with the greatest eponent, write the other variable terms in descending order of their eponents, and put the constant term last. REFLECT 2a. Refer back to the Eplore. Using the tiles, you multiplied 2 by ( + ) and then multiplied 1 by ( + ). You are using the property. 2b. In FOIL, which of the products combine to form the -term? 2c. In FOIL, which of the products combine to form the constant term? 2d. In FOIL, which of the products combine to form the 2 -term? 2e. Two binomials are multiplied to form a trinomial. When is the constant term of the trinomial positive? When is it negative? Chapter 6 351 Lesson 5
3 A-APR.1.1 EXAMPLE Multiplying Two Binomials Using FOIL Multiply (12-5)(3 + 6) using the FOIL method. First Inner (12-5)(3 + 6) = 2 + - - Outer Last (12-5)(3 + 6) = REFLECT 3a. How does the final -term in the answer to the Eample relate to your answer to Question 2b? Eplain. 3b. How does the final constant in the answer to the Eample relate to your answer to Question 2c? Eplain. 3c. How does the final 2 -term in the answer to the Eample relate to your answer to Question 2d? Eplain. 3d. Suppose the problem in the eample were (12-5)(3 + 2). Would the 2 -term in the product change? Would the -term change? Would the constant term change? Eplain your reasoning. 3e. Multiply (12-5)(3 + 2). Chapter 6 352 Lesson 5
As with binomials, you can multiply two polynomials by using the distributive property so that every term in the first factor is multiplied by every term in the second factor. You also use the product of powers property ( a m a n = a m + n ) each time you multiply two terms. 4 A-APR.1.1 EXAMPLE Find the product. Multiplying Polynomials A (4 2 )(2 3-2 + 5) = (4 2 )(2 3 ) + (4 2 )(- 2 ) + (4 2 )(5) Distributive property = 8 5 - + Multiply monomials. B ( - 3)(- 2 + 2 + 1) Method 1: Use a horizontal arrangement. ( - 3)(- 2 + 2 + 1) = (- 2 ) + (2) + (1) - 3(- 2 ) - 3(2) - 3(1) Distribute and then -3. = - 3 + + + - - 3 Multiply monomials. = - 3 + - - Combine like terms. Method 2: Use a vertical arrangement. - 2 + 2 + 1 Write the polynomials vertically. - 3 3 2-6 - 3 Multiply (- 2 + 2 + 1) by -3. - 3 + + Multiply (- 2 + 2 + 1) by. - 3 + - - Add. REFLECT 4a. Is the product of two polynomials always another polynomial? Eplain. 4b. If one polynomial has m terms and the other has n terms, how many terms does the product of the polynomials have before it is simplified? Chapter 6 353 Lesson 5
PRACTICE Find each product. 1. ( + 2)( + 3) 2. ( + 7)( + 11) 3. (2 + 13)( - 6) 4. (2-5)(3 + 1) 5. (2 3 )(2 2-9 + 3) 6. ( + 5)(3 2 - + 1) 7. (2 4-5 2 )(6 + 4 2 ) 8. ( + y)(2 - y) 9. ( + 2y)( 2 + y + y 2 ) 10. ( 3 )( 2-3)(3 + 1) 11. The verte form of a quadratic function is f () = a( - h ) 2 + k. Use your knowledge about multiplying binomials to complete the following. f () = a( - h) ( - ) + k Write as a product of two binomials. = a ( 2 - + 2 ) + k Multiply the binomials. = a2 - + + k Distribute the constant a. Compare this rewritten form to the standard form of a quadratic function, f () = a 2 + b + c. Discuss how b and c relate to the rewritten function. How can you rewrite a quadratic function in verte form so that it is in standard form? 12. The set of polynomials is analogous to a set of numbers you have studied. To determine this set of numbers, consider the following questions about closure. a. Under which operations is the set of polynomials closed? b. Which set of the numbers discussed in the lesson on Rational Eponents is closed under the same set of operations? Chapter 6 354 Lesson 5
Name Class Date 6-5 Additional Practice Multiply. 1. (6m 4 ) (8m 2 ) 2. (5 3 ) (4y 2 ) 3. (10s 5 t)(7st 4 ) 4. 4( 2 5 6) 5. 2(3 4) 6. 7y(3 2 4y 2) 7. ( 3) ( 4) 8. ( 6) ( 6) 9. ( 2) ( 5) 10. (2 5) ( 6) 11. (m 3 3) (5m n) 12. (a 2 b 2 ) (a b) 13. ( 4) ( 2 3 5) 14. (3m 4) (m 2 3m 5) 15. (2 5) (4 2 3 1) 16. The length of a rectangle is 3 inches greater than the width. a. Write a polynomial that represents the area of the rectangle. b. Find the area of the rectangle when the width is 4 inches. 17. The length of a rectangle is 8 centimeters less than 3 times the width. a. Write a polynomial that represents the area of the rectangle. b. Find the area of the rectangle when the width is 10 centimeters. 18. Write a polynomial to represent the volume of the rectangular prism. Chapter 6 355 Lesson 5
Problem Solving 1. A bedroom has a length of 3 feet and a width of 1 feet. Write a polynomial to epress the area of the bedroom. Then calculate the area if 10. 3. Nicholas is determining if he can afford to buy a car. He multiplies the number of months by 30 where represents the monthly cost of insurance, represents the monthly car payment, and represents the number of times he fills the gas tank each month. Write the polynomial that Nicholas can use to determine how much it will cost him to own a car both for one month and for one year. 2. The length of a classroom is 4 feet longer than its width. Write a polynomial to epress the area of the classroom. Then calculate the area if the width is 22 feet. 4. A seat cushion is shaped like a trapezoid. The shorter base of the cushion is 3 inches greater than the height. The longer base is 2 inches shorter than twice the height. Write the polynomial that can be used to find the area of the cushion. (The area of a trapezoid is represented by 1 2 ( 1 2 ).) 5. Which polynomial represents the approimate area of the base of the Great Pyramid? A 90,000 B 2 90,000 C 2 600 90,000 D 2 2 600 90,000 7. The original height of the Great Pyramid was 485 feet. Due to erosion, it is now about 450 feet. Find the approimate volume of the Great Pyramid today. A 562,500 ft 3 C 84,375,000 ft 3 B 616,225 ft 3 D 99,623,042 ft 3 6. Which polynomial represents the approimate volume of the Great Pyramid? F 1 3 3 200 2 30,000 G 1 3 2 200 30,000 H 3 600 2 90,000 J 3 3 600 2 90,000 Chapter 6 356 Lesson 5