1. Multiplying Polynomials -Distributive property -FOIL -Box Method

2. Multiplication of Polynomials Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 To multiply a polynomial by a monomial, use the distributive property and the rule for multiplying exponential expressions. Examples: 1. Multiply: 2x(3x 2 + 2x – 1). = 6x 3 + 4x 2 – 2x 2. Multiply: – 3x 2 y(5x 2 – 2xy + 7y 2 ). = – 3x 2 y(5x 2 ) – 3x 2 y(– 2xy) – 3x 2 y(7y 2 ) = – 15x 4 y + 6x 3 y 2 – 21x 2 y 3 = 2x(3x 2 ) + 2x(2x) + 2x(–1)

3. 3 Multiply: – 2xy(8x 2 +2xy – 5y 2 ). = – 2xy(8x 2 ) – 2xy( 2xy) – 2xy(-5y 2 ) = – 16x 3 y – 6x 2 y 2 + 10xy 3 Multiplying Polynomials-Try it!

4. Try it!. Multiply the polynomial by the monomial. 1) 3(x + 4) 2) 3) Distributive Property

5. FOIL Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 To multiply two binomials use a method called FOIL, which is based on the distributive property. The letters of FOIL stand for First, Outer, Inner, and Last. 1. Multiply the first terms. 3. Multiply the inner terms. 4. Multiply the last terms. 2. Multiply the outer terms. 5. Combine like terms.

6. For use with the product of binomials only! First OuterInnerLast

7. For use with the product of binomials only! First Outer InnerLast

8. For use with the product of binomials only! FirstOuter Inner Last

9. For use with the product of binomials only! FirstOuterInner Last

10. For use with the product of binomials only! FirstOuterInnerLast

11. Example: Multiply Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Examples: 1. Multiply: (2x + 1)(7x – 5). = 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5) = 14x 2 – 10x + 7x – 5 2. Multiply: (5x – 3y)(7x + 6y). = 35x 2 + 30xy – 21yx – 18y 2 = 14x 2 – 3x – 5 = 35x 2 + 9xy – 18y 2 = 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y) First Outer Inner Last First Outer Inner Last

12. Try... FirstOuterInnerLast

13. The third method is the Box Method. This method works for every problem! Here’s how you do it. Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where. This will be modeled in the next problem along with FOIL. 3x-5 5x +2

14. Multiply (3x - 5)(5x + 2) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 15x 2 - 19x – 10 3x-5 5x +2 15x 2 +6x -25x -10 You have 3 techniques. Pick the one you like the best! 15x 2 +6x -25x -10

15. Try it! Multiply (7p - 2)(3p - 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p 2 – 34p + 8 7p-2 3p -4 21p 2 -28p -6p +8 21p 2 -28p -6p +8

16. Multiplication using the Distributive Property Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 To multiply two polynomials, apply the distributive property. Example: Multiply: (x – 1)(2x 2 + 7x + 3). = (x – 1)(2x 2 ) + (x – 1)(7x) + (x – 1)(3) = 2x 3 – 2x 2 + 7x 2 – 7x + 3x – 3 = 2x 3 + 5x 2 – 4x – 3 Two polynomials can also be multiplied using a vertical format. – 2x 2 – 7x – 3 2x 3 + 7x 2 + 3x 2x 3 + 5x 2 – 4x – 3x Multiply – 1(2x 2 + 7x + 3). Multiply x(2x 2 + 7x + 3). Add the terms in each column. 2x 2 + 7x + 3 x – 1 Example:

17. Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x 2 - 5x + 4) - 5(x 2 - 5x + 4) 2x 3 - 10x 2 + 8x - 5x 2 + 25x - 20 Group and combine like terms. 2x 3 - 10x 2 - 5x 2 + 8x + 25x - 20 2x 3 - 15x 2 + 33x - 20

18. x2x2 -5x+4 2x -5 Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. 2x 3 -5x 2 -10x 2 +25x +8x -20 Almost done! Go to the next slide!

19. x2x2 -5x+4 2x -5 Multiply (2x - 5)(x 2 - 5x + 4) Combine like terms! 2x 3 -5x 2 -10x 2 +25x +8x -20 2x 3 – 15x 2 + 33x - 20

20. Example: Word Problem Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Example: The length of a rectangle is (x + 5) ft. The width is (x – 6) ft. Find the area of the rectangle in terms of the variable x. A = L · W = Area x – 6 x + 5 L = (x + 5) ft W = (x – 6) ft A = (x + 5)(x – 6 ) = x 2 – 6x + 5x – 30 = x 2 – x – 30 The area is (x 2 – x – 30) ft 2.