14-6

1. Multiplying Binomials 14-6 Warm Up Problem of the Day Lesson Presentation Course 3

2. Warm Up Multiply. 1.x(x + 2) 2.–3(p –4) 3. 2x(3x– 7) 4. Find the area of a rectangle with length 16 cm and width 21 cm. x2 + 2x –3p + 12 6x2– 14x 336 cm2 Course 3

3. Problem of the Day Find the missing term in the given arithmetic sequence. 170, , , , 140,. . . 162.5, 155, 147.5

4. Learn to multiply binomials.

5. Vocabulary FOIL

6. You can use the Distributive Property to multiply two binomials. (x + y)(x +z) = x(x + z) + y(x + z) = x2 + xz + xy + yz

7. 2 (x + y)(x + z) = x xy yz xz + + + The product can be written as FOIL: The First terms, the Outer terms, the Inner terms, and the Last terms of the binomials. First Last ( X + Y)(X + Z) = X Inner Outer

8. Additional Example 1: Multiplying Two Binomials Multiply. B. (x + 3)(x + z) A. (n – 2)(m– 8) (n – 2)(m – 8) Foil (x + 3)(x + z) x2 + xz + 3x + 3z mn – 8n – 2m + 16

9. Additional Example 1: Multiplying Two Binomials Multiply. D. (x– 4y)(x + 7y) C. (a + 3)(a + 5) (a + 3)(a + 5) Foil (x – 4y)(x + 7y) a2 + 5a + 3a + 15 x2 + 7xy – 4xy – 28y2 x2 + 3xy – 28y2 a2 + 8a + 15 Combine like terms.

10. Helpful Hint When you multiply two binomials, you will always get four products. Then combine like terms.

11. Check It Out: Example 1 Multiply. B. (y + 2)(y + r) A. (r – 4)(s– 6) (r – 4)(s – 6) Foil (y + 2)(y + r) y2 + ry + 2y + 2r rs – 6r – 4s + 24

12. Check It Out: Example 1 Multiply. D. (x– 3y)(x + 8y) C. (a + 3)(a + 4) (a + 3)(a + 4) Foil (x – 3y)(x + 8y) a2 + 4a + 3a + 12 x2 + 8xy – 3xy – 24y2 x2 + 5xy – 24y2 a2 + 7a + 12 Combine like terms.

13. Additional Example 2 An 8 in. by 10 in. photo has a wooden frame that extends x in. beyond each side. Represent the area of the frame in terms of x. Base: 8 + 2x Height: 10 + 2x Area of the photo and frame combined: A = (8 + 2x)(10 + 2x) = 80 + 16x + 20x + 4x2 = 80 + 36x + 4x2

14. Additional Example 2 Continued The photo area is 8 · 10 = 80 in2, so the frame area is 80 + 36x + 4x2 – 80 = 36x + 4x2 in2.

15. Check It Out: Example 2 Find the area of the border of a computer screen of width x centimeters around a 50 cm by 80 cm screen. Represent the area of the border in terms of x. Base: 80 + 2x Height: 50 + 2x Area of the screen and border combined: A = (80 + 2x)(50 + 2x) = 4000 + 160x + 100x + 4x2 = 4000 + 260x + 4x2

16. Check It Out: Example 2 Continued The screen area is 80 · 50 = 4000 cm2, so the frame area is 4000 + 260x + 4x2 – 4000 = 260x + 4x2 cm2.

17. Binomial products of the form (a + b)2, (a –b)2, and (a + b)(a – b) are often called special products.

18. (n – m) (n – m) Additional Example 3: Special Products of Binomials Multiply. B. (n–m)2 A. (x + 6)2 (x + 6)2 (n – m)2 (x + 6)(x + 6) x2 + 6x + 6x + 36 n2 – nm – nm + m2 x2 + 12x + 36 n2 – 2nm + m2

19. Additional Example 3: Special Products of Binomials Multiply. C. (x – 7)(x + 7) (x – 7)(x + 7) x2 + 7x – 7x – 49 x2 – 49

20. (r – 3) (r – 3) Check It Out: Example 3 Multiply. B. (r–3)2 A. (x + y)2 (x + y)2 (r – 3)2 (x + y)(x + y) x2 + xy + xy + y2 r2 – 3r – 3r + 9 x2 + 2xy + y2 r2 – 6r + 9

21. Check It Out: Example 3C Multiply. C. (x –y)(x +y) (x – y)(x + y) x2 + xy – xy – y2 x2 – y2

22. Special Products of Binomials (a + b)2 = a2 + ab + ab + b2 = a2 + 2ab + b2 (a – b)2 = a2 – ab – ab + b2 = a2 – 2ab + b2 (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2

23. Lesson Quiz: Part 1 Multiply. 1. (x + 5)(x + 2) 2. (c – 4)(c – 2) 3. (2y + 2)(y – 3) 4. (2m + 2)(3n + 2) x2 + 7x + 10 c2 – 6c + 8 2y2 – 4y – 6 6mn + 4m + 6n + 4

24. Lesson Quiz: Part 2 Multiply. 5. (a – 1)2 6. (b + 3)(b – 3) 7. Find the area of a cement walkway of width w yd that is around a 20 yd by 30 yd pool. a2 – 2a + 1 b2 – 9 4w2 + 100w yd2