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Factoring

Factoring. MM1A2 Students will simplify and operate with radical expressions, polynomials, and rational expressions. f. Factor expressions by greatest common factor, grouping, trial and error, and special products. Warm up. Complete the handouts on factor tables…. What are Factors?.

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Factoring

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  1. Factoring MM1A2 Students will simplify and operate with radical expressions, polynomials, and rational expressions. f. Factor expressions by greatest common factor, grouping, trial and error, and special products.

  2. Warm up • Complete the handouts on factor tables…

  3. What are Factors? • The expressions that are multiplied together to get a polynomial • What are the factors of 10? • What about the factors of 39? • Now, what about the factors of 2x + 4?

  4. Greatest Common Factor • Determine the greatest number that can be divided into each term of an expression • Analyze each term to see what the factors of that term are • Find the common factor for all terms • “Pull” that common factor out of the expression • Write it with the factors outside of the parentheses

  5. GCF with Binomials • Find the GCF of 2x2 + 6x • What are the factors of 2x2? 2 & x • What are the factors of 6x? 2, 3, and x • What are their common factors? 2 and x • Now, “pull” 2x out of each term and write the “leftovers” in parentheses • 2x (x + 3)

  6. Ok…try these… • 16m – 8 • 120t2 – 140 • 34 + 17y2

  7. Finding the GCF with Trinomials • Using the same procedure as binomials, we look for the GCF in the trinomial • For example: 4x2 + 16x + 32 • What is the GCF? 4 will divide into each term • Rewrite the expression with the GCF of 4 “pulled” from it • 4(x2 + 4x + 8)

  8. Let’s Try A Few More!!  • 9c2 + 24c + 30 • - 4w2 + 100 – 20w • 3m2 + 30m + 75

  9. Factoring Trinomials • Trinomials in the form of x2 + bx + c • Use the following to factor trinomials in this form. (x2 + bx + c) = (x + p)(x + q) • Use this as long as p + q = b and pq = c

  10. Factoring when b and c are positive • x2 + 10x + 24 • Find two positive factors of 24 whose sum is 10. • Make an organized list

  11. What are the factors of 24?

  12. Which factors give you a sum of 10? • Now, take those factors and put them into the (x + p)(x + q) form • You would have (x + 6)(x + 4) • These are your factors for x2 + 10x + 24

  13. Check… • Using FOIL, multiply your binomials out to see if you come up with the same trinomial • Did it work?

  14. Try these… • b2 + 8b + 7 • p2 + 10p + 25 (hint…there are 2 ways to write this one!) • Don’t forget to check!!

  15. Factoring when b is negative and c is positive • Think about what happens when you have a positive and a negative number multiplied together… • When your “b” value is negative, that means the numbers in the factors are also both negative • What happens when you multiply a negative times a negative? • Your “c” value will be positive since you are multiplying negative numbers together

  16. w2 - 10w + 9 • What are the factors of 9? Organize them… • Now, add the factors up to find their sum • Be careful of those negatives!!! Now, which have a sum to be -10 (the “b” value)? So, the factors of the expression above are (w – 9)(w – 1)

  17. Try these… • b2 – 7b + 10 • m2 – 10m + 24

  18. Factoring when b is positive and c is negative • If the “c” value is negative, that means that the two factors have different signs • Why is this true? • Negative times a positive will yield a negative! • Use the same procedure with finding the numbers that are the factors of the “c” value and finding the sum of those factors • Then, you use the factors that add up to be the “b” value!

  19. Factor k2 + 6k - 7 • What are the factors of 7? -7, and 1 7, and -1 • What are the sum of those factors? -7 + 1 = -6 7 + (-1) = 6 • Look back at your problem… we have a positive 6 in the “b” position • So, what are your factors of this trinomial? (k + 7)(k – 1) = k2 + 6k - 7

  20. Try these… • n2 + 2n – 48 • y2 – 5y - 24

  21. Guided Practice • x2 + 10x + 16 • y2 + 6y + 5 • z2 – 7z + 12 • x2 + 10x – 11 • y2 + 2y – 63 • z2 – 5z - 36

  22. Complete Factoring Practice #1

  23. Warm up (ti) • X2 – x – 6 • X2 - 3x + 10 • X2 – 8x – 9 • X2 + 3x + 10

  24. Factoring Special Products • Difference of Two Squares a2 – b2 = (a + b)(a – b) • Perfect Square Trinomial Pattern a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2

  25. Let’s look at the difference of two squares pattern… • a2 – b2 = (a + b)(a – b) • First, using FOIL multiply out the (a + b)(a – b) • What happens to the “b” value? • This is the difference of two squares pattern • You must learn to recognize it… it makes it a lot easier when you spot it!

  26. Any time you see a squared value with a subtraction sign and another squared value, you are looking at a difference of two squares pattern • Let’s try this… x2 – 25 • What are your factors of -25? • 5 and -5 • Put these in factored form (x + 5)(x – 5) • FOIL just to check that the “b” value is gone!

  27. Your Turn… • m2 – 121 • z2 - 49

  28. What if I have 2 different variables?? • No problem… treat them as part of the factors! • s2 – 4t2 • What factors will give me -4t2 but will leave me without a “b” value? -2t, and 2t • Using this information, list the factors (s – 2t)(s + 2t)

  29. Ok…your turn… • y2 – 64z2

  30. Perfect Square Trinomial Patterns • a2 + 2ab + b2 and a2 – 2ab + b2 • Let’s start with the first one… a2 + 2ab + b2 • Teacher demonstration on board • x2 + 6x + 9

  31. Try this one… • X2 + 14xy + 49y²

  32. What about a2 – 2ab + b2?? • Teacher demonstration on board • x2 – 12x + 36

  33. Try this one… • x2 – 16x + 64

  34. Guided Practice • c2 – 144 • r2 + 14r + 49 • x2 – 18xy + 81y2 • s2 – 16m2 • m2 – 1/2m + 1/16

  35. Solving Equations with Special Products • Solve the equation q2 – 100 = 0 • Write the left side as the difference of two squares q2 – 102 = 0 • Factor out the left side using the difference of two squares pattern (q – 10)(q + 10) • Set each factor equal to zero and solve for the variable • q = 10 and -10

  36. Try this one… • r2 – 10r + 25 = 0

  37. Complete Special Products Practice

  38. Sum and Difference of Cubes • x³ + y³ = (x + y)(x² - xy + y²) • x³ - y³ = (x - y)(x² + xy + y²) • Example : x³ - 8 • (x – 2)(x² + 2x + 4)

  39. Example x³ + 27 (x + 3)(x2 – 3x + 9)

  40. Example 8x3 – 64

  41. Practice • Factor x3y6 – 64 • (xy2 – 4)(x2y4 + 4xy2 + 16)

  42. Warm up • Factor completely. • x3- 8 • 27x3 + y3 • X4 + x • X3 + 64 y3

  43. Solving Polynomial Equations using Grouping • Rearrange your expression where you can group terms together to later take out the GCF • For example: m3 + 7m2 – 2m – 14 • We group: (m3 + 7m2) + (-2m – 14) • Now, factor each grouping using the distributive property m2 (m + 7) + -2 (m + 7)

  44. Grouping, cont’d. • What do you notice about the factors? • They are sharing the same terms inside of the parentheses • Now, rewrite your factors—group together the GCFs (m + 7) (m2 – 2)

  45. Let’s Practice—Factor by Grouping • n3 + 30 + 6n2 + 5n • x3 + 3x2 + 2x+ 6

  46. Practice worksheet on grouping

  47. Solving Polynomial Equations • To solve the equation h2 – 4h = 21… - Write your original equation • Subtract 21 from each side so that you have a trinomial on one side and zero on the other • Factor the left side • Now, using the zero-product property, solve for the variable. • The solutions are the answers to the above step!

  48. h2 -4h = 21 • Subtract 21 from both sides h2 – 4h – 21 = 21 – 21 • Simplified would give you h2 – 4h – 21 = 0 • Factor the left side • What factors of 21 when added will give you -4 as their sum? • -7 and 3 OR 7 and -3?

  49. Now, using those two numbers, put them in the factored form for the equation (x – 7)(x + 3) • Using the zero-product property, set each of the factors equal to zero and solve for the variable—these are your solutions! • h + 3 = 0 h = -3 • h – 7 = 0 h = 7

  50. Your Turn… • X2 + 30 = 11x • X2 + 7x + 12 = 0 • a2 – 17a + 30 = 0 • 2x3 – 14x2 + 12x = 0 • w2 – 81 = 0 • m2 – 7m = 0 • 4y2 - 49 = 0

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