1. POLYNOMIALS

2. MULTIPLYING POLYNOMIALS

3. REVIEW Polynomials:

4. To multiply polynomials: 1)Multiply each term in one polynomial by each term in another polynomial 2)Simplify as needed (collect like terms)

5. 1 TERM X 1 TERM Multiplying a monomial by another monomial: 1) Multiply the constants together 2) Multiply the variables together 3) Combine the result

6. EXAMPLE: (2y)(3y) 1) (2)(3) = 6 2) (y)(y) = ? NOTE: We use brackets to represent multiplication so that we don’t confuse it with our variable ‘x’ 1) Multiply the constants together 2) Multiply the variables together 3) Combine the result

7. MULTIPLYING VARIABLES WITH EXPONENTS We know y² = yy We know y³ = yyy So (y²)(y³) = yyyyy = y ⁵ However, instead of breaking it down into y’s, we can just add the exponents: So (y²)(y³) = y² ⁺ ³ = y ⁵ Let’s look back at our example

8. EXAMPLE: (2y)(3y) 1) (2)(3) = 6 2) (y)(y) = ? (y)(y) = y¹ ⁺ ¹ = y² 3) (2y)(3y) = 6y² 1) Multiply the constants together 2) Multiply the variables together 3) Combine the result

9. EXAMPLE: (4z²)(1/2z³) 1) Multiply the constants together 2) Multiply the variables together 3) Combine the result

10. EXAMPLE: (-6x ⁴ y)(2xz³) 1) Multiply the constants together 2) Multiply the variables together 3) Combine the result

11. EXAMPLE: (-3ab ⁴ c²)(-4a²bc³) 1) Multiply the constants together 2) Multiply the variables together 3) Combine the result

12. 1 TERM X 2 TERMS We call this the Distributive Property Multiply the single term by each of the two terms

13. LET’S CHECK IF IT WORKS

14. EXAMPLE: 2(x+7)

15. EXAMPLE: 2x(x³+3y)

16. EXAMPLE: (b³-2abc) 3a²b

17. 1 TERM X 3+ TERMS This is still the distributive property We will multiply the single term by each of the other terms a(b + c + d + …) = ab + ac + ad + …

18. EXAMPLE 4y²(3x ⁵ - xy³ - y²z)

19. EXAMPLE (5a ⁵ - b³ + 4a²b) 2b²

20. 2 TERMS X 2 TERMS (a + b)(c + d) = ac + ad + bc + bd

21. This process is called FOILING First Outside Inside Last

23. STEP BY STEP

24. EXAMPLE:

25. (x – 2)(x+6)

26. EXAMPLE: (4x – 2)(3x+6)

27. EXAMPLE: (xy – 2y)(3x²+6)

28. EXAMPLE: (3x³y + 2y³)(x² - 4xy)

29. 2 TERMS X 3+ TERMS We can extend the FOIL method to any polynomials being multiplied together (a + b)(c + d + e) = ac + ad + ae + bc + bd + be Multiply ‘a’ with all the terms in the second polynomial Multiply ‘b’ with all the terms in the second polynomial

30. ANY TERMS X ANY TERMS This would be true for any number of terms (a + b + c)(d + e + f) = ad + ae + af + bd + be + bf + cd + ce + cf

31. EXAMPLE: (x + 2y)(3x − 4y + 5)

32. EXAMPLE: (x - 4)(3x - y + 3)

33. EXAMPLE: What is the product of (2y - 1), (2y + 1) and (4y 2 + 1)?

34. YOUR TURN!!