1. Extending the Distributive Property

2. You already know the Distributive Property …

3. So far you have used it in problems like this:

4. The distributive property is used all the time with polynomials.  One thing it lets us do is multiply a monomial times a larger polynomial.

5. You normally wouldn’t show the work, but this is the distributive property.

6. You just take the monomial times each of the terms of the polynomial, one at a time. So 3x 2 (5x 2 – 2x + 3) = 15x 4 – 6x 3 + 9x 2

7. When you multiply each term, it’s the basic rules of multiplying monomials.  Multiply the coefficients.  Add the exponents.

8. Multiply: 2n 5 (3n 3 + 5n 2 – 8n – 3) 8x 4 y 3 (2x 2 y 2 + 7x 5 y)

9. Multiply: 2n 5 (3n 3 + 5n 2 – 8n – 3) 6n 8 + 10n 7 – 16n 6 – 6n 5 8x 4 y 3 (2x 2 y 2 + 7x 5 y) 16x 6 y 5 + 56x 9 y 4

10. Multiply: -9m(2m 2 – 7m + 1) 4x 2 y(3x 2 – 4xy 4 + 2y 5 )

11. Multiply: -9m(2m 2 – 7m + 1) -18m 3 + 63m 2 – 9m 4x 2 y(3x 2 – 4xy 4 + 2y 5 ) 12x 4 y – 16x 3 y 5 + 8x 2 y 6

12. You can extend the distributive property to multiply two binomials, like (x + 2)(x + 3) or(3n 2 + 5)(2n 2 – 9)

13. To multiply essentially you distribute the “x” and then distribute the “2”

14. To multiply essentially you distribute the “x” and then distribute the “2” x 2 + 3x

15. To multiply essentially you distribute the “x” and then distribute the “2” x 2 + 3x + 2x + 6

16. x 2 + 3x + 2x + 6 To finish it off, you combine the like terms in the middle.

17. x 2 + 3x + 2x + 6 To finish it off, you combine the like terms in the middle. 5x The final answer is x 2 + 5x + 6

18. (3n 2 + 5)(2n 2 – 9)

19. (3n 2 + 5)(2n 2 – 9) Distribute 3n 2 – then distribute 5 6n 4 – 27n 2 + 10n 2 – 45

20. (3n 2 + 5)(2n 2 – 9) Distribute 3n 2 – then distribute 5 6n 4 – 27n 2 + 10n 2 – 45 Combine like terms -17n 2

21. (3n 2 + 5)(2n 2 – 9) Distribute 3n 2 – then distribute 5 6n 4 – 27n 2 + 10n 2 – 45 Combine like terms -17n 2 6n 4 – 17n 2 – 45

22. There are lots of ways to remember how the distributive property works with binomials. x 2 + 6x + 4x + 10 = x 2 + 10x + 24

23. The most common mnemonic is called F O I L

25. In Gaelic, FOIL is CAID.

26. However you remember it, it’s just the distributive property.

27. Multiply (3x – 5)(2x + 3) (x 3 + 7)(x 3 – 4)

28. Multiply (3x – 5)(2x + 3) 6x 2 + 9x – 10x – 15 = 6x 2 – x – 15 (x 3 + 7)(x 3 – 4) x 6 – 4x 3 + 7x 3 – 28 = x 6 + 3x 3 – 28

29. Multiply (2n – 5)(3n – 6) (x + 8)(x – 8)

30. Multiply (2n – 5)(3n – 6) 6n 2 – 12n – 15n + 30 = 6n 2 – 27n + 30 (x + 8)(x – 8) x 2 – 8x + 8x – 64 = x 2 – 64

31. Now consider (2x 5 + 3) 2 and (n – 6) 2

32. Now consider (2x 5 + 3) 2 and (n – 6) 2 This just means (2x 5 + 3)(2x 5 + 3) and (n – 6)(n – 6)

33. (2x 5 + 3) 2 (2x 5 + 3)(2x 5 + 3) 4x 10 + 6x 5 + 6x 5 + 9 4x 10 + 12x 5 + 9

34. (n – 6) 2 (n – 6)(n – 6) n 2 – 6n – 6n + 36 n 2 – 12n + 36

35. Multiply (x + 4) 2 (p 3 – 9) 2

36. Multiply (x + 4) 2 = x 2 + 8x + 16 (p 3 – 9) 2 = p 6 – 18p 3 + 81

37. You can extend the distributive property even further … Multiply (3g – 3)(2g 2 + 4g – 4)

38. Multiply (3g – 3)(2g 2 + 4g – 4)

39. Multiply (x 2 + 5)(x 2 – 11x + 6)

41. CHALLENGE: Multiply (2x 2 + x – 3)(x 2 – 2x + 5)