March 12, 2009 Factoring page 1 Factoring Last class, you worked on a set of problems where you had to do multiplication table calculations in reverse. For example, given the answer x 2 + 4x + 2x + 8, you could fill in the outside of the multiplication table as shown here: x 4 x x 2 4x 2 2x 8 That would tell you that x 2 + 4x + 2x + 8 = (x + 2)(x + 4). This procedure is called factoring because you have found the two factors (x + 2) and (x + 4) that multiply to x 2 + 4x + 2x + 8. However, sometimes factoring is harder because the answer has already been simplified. For example, suppose you were given the simplified answer x 2 + 6x + 8. To use the multiplication table in reverse you d have to figure out that the 6x came from having 4x + 2x before simplifying. So generally, to reverse the Distributive Property working from a simplified answer, it is necessary to figure out how to split up the combined term into its original parts. During this lesson you ll learn two methods for doing this: Trial-and-error Using a pattern involving addition and multiplication Method 1: Trial-and-error Example: Factor x 2 + 6x + 8. Solution: Right away we can see that x 2 + 6x + 8 could be the answer of a two-by-two multiplication table that looks like this: x x 2? 8 The blank inside spaces have to add up to 6x, but there are a few different ways this could happen: x x 2 3x? 3x 8 x x 2 4x? 2x 8 x x 2 5x? 1x 8 x x 2 6x? 0x 8 If you try to find the? numbers in all four possible tables, you will only be successful with one of them. The successful table here is: x 4 x x 2 4x 2 2x 8 The table gives this equation as an answer: x 2 + 6x + 8 = (x + 2) (x + 4). You try it 1. Factor using the trial and error method. Fill in a multiplication table, then write an equation.
March 12, 2009 Factoring page 2 a. Factor x 2 + 6x + 5. b. Factor x 2 + 6x + 9. c. Factor x 2 + 5x + 4. d. Factor x 2 + 8x + 12.
March 12, 2009 Factoring page 3 e. Factor x 2 + 8x + 16. f. Factor x 2 + 8x + 15. g. Factor x 2 + 9x + 8. h. Factor x 2 + 2x + 1.
March 12, 2009 Factoring page 4 Method 2: Addition-and-multiplication A pattern that you might have noticed in the problems that you ve done so far is that you need to think about both addition and multiplication. For example, the answer x 2 + 6x + 8 = (x + 2) (x + 4) depends on the facts 6 = 2 + 4 and 8 = 2 4. This suggests the following method: Addition-and-multiplication method: To factor x 2 + bx + c, you need to think of two numbers that when added together make b, and when multiplied together make c. The numbers that you think of become the outside numbers of the multiplication table. Example: Factor x 2 + 8x + 12. Solution: We need to think of two numbers that add up to 8 and multiply to 12. The numbers 2 and 6 work: 2 + 6 = 8 and 2 6 = 12. These give us the multiplication table: x 6 x x 2 6x 2 2x 12 We get this equation as the answer: x 2 + 8x + 12 = (x + 2) (x + 6). You try it 2. Factor using the addition-and-multiplication method. (If you get stuck finding the pair of numbers, try one of the other methods as a backup.) Write your answer both as a multiplication table and as an equation. a. Factor x 2 + 12x + 35. b. Factor x 2 + 12x + 36. c. Factor x 2 + 12x + 27.
March 12, 2009 Factoring page 5 d. Factor x 2 + 13x + 12. e. Factor x 2 + 14x + 48. f. Factor x 2 + 13x + 36. g. Factor x 2 + 502x + 1000. h. Factor x 2 + 2000x + 1,000,000.
March 12, 2009 Factoring page 6 When there are negative numbers When you have a factoring problem involving any negative numbers: The trial-and-error method becomes harder because there are more numbers to try, but you might still get it to work. The addition-and-multiplication method usually works the best. See the example below. Example: Factor x 2 2x 8. Solution: We need to think of two numbers that add up to 2 and multiply to 8. The numbers 4 and 2 work: 4 + 2 = 2 and 4 2 = 8. These give us the multiplication table: x 2 x x 2 2x 4 4x 8 We get this equation as the answer: x 2 2x 8 = (x 4) (x + 2). You try it 3. Factor using the addition and multiplication method. Write your answer both as a multiplication table and as an equation. a. Factor x 2 + 2x 35. b. Factor x 2 12x + 35. c. Factor x 2 6x + 8.
March 12, 2009 Factoring page 7 d. Factor x 2 + 2x 15. e. Factor x 2 2x 15. f. Factor x 2 8x + 15. g. Factor x 2 + 6x 7. h. Factor x 2 6x 7. i. Factor x 2 16x + 64.
March 12, 2009 Factoring page 8 j. Factor x 2 16x + 60. k. Factor x 2 + 4x 60. l. Factor x 2 + x 6. m. Factor x 2 x 6. n. Factor x 2 4. Hint: Think of it as x 2 + 0x 4. o. Factor x 2 81.