Factorization Introduction
- 1. Expanding and Factoring Polynomial Expressions { Different ways to look at things.
- 2. Polynomial Expressions Expression—Symbols with meaning Exs: 1 + 2 3a2 - b sin(π)
- 3. Polynomial Expressions Polynomial Expression— Expression with these symbols: + - x variables coefficients exponents (limited)
- 4. Polynomial Expressions Which is not one? a. 1 + 2 b. 3a2 - b c. sin(π)
- 5. Polynomial Expressions Which is not one? a. x2 + y2 b. 3a/b c. 0
- 6. Polynomial Expressions Which is not one? a. 3a + 2b +4c b. 4a-2.5 + b c. 92 + 8m
- 7. Binomial Expansion Binomial—Polynomial with two terms Exs: 3a + b 3x2 + 4
- 8. Binomial Expansion Expansion? Exponentially increasing a Binomial See the following: (x + 4)2
- 9. Binomial Expansion How do we expand (x + 4)2? x2 + 42 = x2 + 16? No!
- 10. FOIL Method How do we expand (x + 4)2? We use the FOIL method, an acronym for distribution.
- 11. FOIL Method First Outer Inner Last
- 12. FOIL Method (x + 4)2 = (x + 4)(x + 4) F (x + 4)(x + 4) » x2 O (x + 4)(x + 4) » 4x I (x + 4)(x + 4) » 4x L (x + 4)(x + 4) » 16 = x2 + 4x + 4x + 16 = x2 + 8x + 16
- 13. FOIL Method Applies to any multiplication of binomials. Ex: (x + 2)(3x - 7) Example in Action
- 14. FOIL Method Practice Problems: 1. (x - 11)2 2. (2x - 4)(2x + 3) 3. (a + b)2 4. (a + b)(c + d)
- 15. Factorization How do you reverse foil? A process called factoring!
- 16. Factorization When factoring, you are a detective. Think about the clues.
- 17. Factorization x2 + 7x + 10 What are the clues? Recall FOIL and reverse it.
- 18. Factorization x2 + 7x + 10 F: the first term is the product of the two first terms. What gives us x2?
- 19. Factorization x2 + 7x + 10 F: x and x L: the last term is the product of two terms. What gives us 10?
- 20. Factorization x2 + 7x + 10 F: x and x L: 10 and 1, 5 and 2, -10 and -1, OR -5 and -2. OI: the middle term is the sum of the outer and inner products. What gives us 7?
- 21. Factorization x2 + 7x + 10 F: x and x L: 10 and 1, 5 and 2, -10 and -1, OR -5 and -2. OI: 5 and 2 So, our answer is (x + 5)(x +2).
- 22. Factorization Use the FOIL method to check your answer: (x + 5)(x + 2) The result is x2 + 7x + 10 just as it should be.
- 23. Factorization Another example question: x2-2x-24
- 24. Factorization x2 - 2x - 24 Use the clues: F: What produces x2? x and x
- 25. Factorization x2 - 2x - 24 Use the clues: L: What produces -24? -24 and 1, -1 and 24, -12 and 2, -2 and 12, -8 and 3, -3 and 8, -6 and 4, OR -4 and 6
- 26. Factorization x2 - 2x - 24 Use the clues: OI: What makes -2? -6 and 4
- 27. Factorization x2 - 2x - 24 Putting the clues together makes the answer: (x - 6)(x + 4)
- 28. Factorization Why? Why factor? When will you need this?
- 29. Solutions! Why Factor? • It’s fun to be a detective. • It makes things look pretty. • It gives solutions!
- 30. Solutions! How does it give solutions? x2 - 2x – 24 = 0 (x – 6)(x + 4) = 0 x = 6, x = -4