This Factoring Packet was made possible by a GRCC Faculty Excellence grant by Neesha Patel and Adrienne Palmer. FACTORING HANDOUT A General Factoring Strategy It is important to be able to recognize the different types of polynomials and know each ones factoring method. Use these steps to guide you: 1) Factor out the greatest common factor (GCF),if there is one. ) Are there two terms? (Binomial) Is it Difference of two squares? If yes, factor by using: Page Note: You cannot factor a binomial in the form. 3) Are there three terms? (Trinomial) Is it a perfect square trinomial? If yes use: Page 3 4) Is the form where? Factor by Product-Sum Method. (Page 5) 5) Is the form where? Factor by Guess and Check, the ac-grouping or one of the other methods attached. (Page 6 and 7) 6) If you can t factor it by any method above, the polynomial is irreducible. It is prime. 1
Factoring Binomials Difference of Two Squares: Example: Factor *Notice that both and 9 are perfect squares: and So Example: Factor Factor out the GCF first: *Notice that both and are perfect squares: and So
Factoring Trinomials Perfect Square Trinomials: Example: Factor *Notice that both and 9 are perfect squares. So a good first guess at how to factor this trinomial would be to use their roots: Then we can just work on figuring out what signs need to go in each parentheses. With a little trial and error, we see that a minus sign in each parentheses would work. So Another way that we could have looked at this factoring problem would be to notice that and and [if we are trying to match things up with the special factoring patterns for perfect square trinomials, then ]. Recognizing the special factoring pattern, we could have factored immediately into the form Example: Factor *Notice that both and 5 are perfect squares. So a good first guess at how to factor this trinomial would be to use their roots: Then we can just work on figuring out what signs need to go in each parentheses. With a little trial and error, we see that a plus sign in each parentheses would work. So 3
Another way that we could have looked at this factoring problem would be to notice that and and [if we are trying to match things up with the special factoring patterns for perfect square trinomials, then ]. Recognizing the special factoring pattern, we could have factored immediately into the form Example: Factor *With a little practice, you may notice that So using the special factoring pattern we get 4
FACTORING TRINOMIALS OF THE FORM ax bx c USING THE PRODUCT-SUM METHOD (Use when a=1) Example: x 5x 6 STEPS 1. Setup the binomial factors and enter the first term of each factor. Remember, you re doing the reverse of FOILING. (x )(x ). Write the value of b and c : b = 5, c = 6 3. List all pairs of integers whose product is c. C= 6 1 6 3 4. Choose the pair whose sum is b: b = 5 (This one) 5. Plug the matching pair into the binomial factors: ( x )( x 3) 5
FACTORING TRINOMIALS IN THE FORM FACTOR BY GUESS AND CHECK Use this method if the a and c values are small or prime. E.g. Factor Think reverse FOILing. The only choice for the first terms in each binomial is 5x and x to obtain the product of that appears in the first term above. We wish to obtain the c value of when we FOIL back. Our factors of are 1 and. So either we have: Or FOILing the first option gives the middle term of that appears in the original trinomial. 6
Use when or 0: Example 1: FACTORING TRINOMIALS IN THE FORM FACTOR BY THE ac AND GROUPING METHOD STEPS: 1. Factor out a GCF if there is one. This example does not have one.. Then use the steps below to factor the trinomial into two binomial factors. 3. List the values of a, b and c in the expression: 4. Find the product of : 5. List the factor pairs that give the product of : 6. Find the pair of factors whose sum equals b, and write as (i.e. The middle term including the variable) 7. Replace these two terms for bx in the original expression, so that you now have an expression with 4 terms: 8. Use the Grouping Method to complete the factoring as follows: Group the first two terms together and the last two terms together: 9. Factor out any common factors from the first group and any from the second group: Ist term nd term Notice that we now have an expression with just terms. Each term should have a common factor (x + 1 in this case). 10. Factor out this common factor from each term:. These are your binomial factors. 11. FOIL out to double check that your factors match the original equation. 7
FACTOR BY THE ac AND GROUPING METHOD continued: Example : 1. Factor out the GCF:. Ignore the GCF for now. We wish to factor. List the values of a, b and c for this quadratic expression: a =, b = -15, c = -7 3. Find the product of a c : 4. List all factor pairs of the product in step 3. Be systematic and keep going until you find the pair whose sum equals (the value of b). Notice that the signs of the factors must be opposites: for example. The two factors we want are 3 and Rewrite the middle term using these factors: 5. Replace these two terms for bx in the original expression, so that you now have an expression with 4 terms: 6. Group the first two terms together and the last two terms together: 7. Factor the GCF from each set of parentheses: Notice that we now have an expression with just two terms. Each term has a common factor of. 1 st term nd term 8. Factor out this common factor from each term to obtain your two binomial factors: Note: If you had reversed the two middle terms in step 5 to obtain Be careful how you handle the parentheses. If you have a minus outside the second set of parentheses, you will need to change the sign of every term inside the parentheses as follows: Both the 18x and 7 change signs. 8
FACTORING TRINOMIALS IN THE FORM ax bx c USING THE TABLE METHOD Use when a 1 or 0: STEPS: 1. Example: x 7x 3. Factor out a GCF if there is one. Then use the steps below to factor the remaining trinomial. 3. List the values of a, b and c in the quadratic expression: a =, b = 7, c = 3 Setup a box as shown below. Write the value of unshaded box. in the top left unshaded box and c bottom right 3. Above example Factor Factor First term ax Last term c x 3 5. Find the product of ac : 3 6 6. List factors of the product in step 5: 1 6, 3 7. Find the pair of factors whose sum equals b, and write as 1 x 6 x (i.e. The middle term, i.e. include the variable) 8. Plug these two terms into the two unshaded empty boxes in the table. (it doesn t matter which term goes into which box). Then factor out the common factors in each row and column and place these in the shaded boxes: 9
Factor x 3 x x 6x 1 x 3 9. The two shaded boxes give you the factored binomials products: ( x 1)( x 3) FOIL out to check you get the original trinomial expression. 10
Practice Problems Begin by doing all of the problems that end with a 7 (problems 7, 17, 7, etc.). Check your answers by multiplication; if you multiply your answer out and simplify, you should get the original polynomial. For problems that you have trouble with, work on the other nine problems in that group. Again, check your answers by multiplication. Group A 1. x 7x 10. x 9x 8 3. n 7n 1 4. a 11a 30 5. z 10z 4 6. t 8t 1 7. x 9x 18 8. x 15x 36 9. x 11x 18 10. m 13m 30 Group B 16. m 5m 36 17. x 5x 4 18. a a 4 19. x x 4 0. t t 0 Group C 1. 3x 5x. 5y 1y 4 3. a 9a 4 4. 3n 10n 3 5. 4z 9z 5 Group D 31. x x 3 3. 3t t 33. 4t 8t 5 34. 4m 3mn n 35. 9q 6q 8 36. 6w 5w 6 37. 3n 4n 4 38. 4x 7xy y 39. 3y y 10 40. 5t 3t 6. 4x 7xy 3y 11. x x 1 1. y 4y 3 13. z z 15 7. x 11x 5 8. 9t 9t 9. 4w 8w 3 14. x 11xy 1y 30. 7x 19x 10 15. n n 8 11
Group E 73. 36y 84yz 49z 41. 1x 31x 0 4. 18y 7yz 10z 43. 0a 56a 15 44. 4t 6t 35 45. 36n 89n 18 46. 6x 5x 1 47. 6y 9y 0 48. 1y 8yx 15x 49. 0m 39m 18 50. 30x 109x 30 Group F 58. 1c 31c 30 59. 40a 3ab 30b 60. 1x 19x 10 Group G 61. b 9 6. 4z 81 63. 36s 11t 64. 144x 5 65. 50x 3 66. 9a 64 67. 16x 49y 68. 81n 100 74. 9a 48a 64 75. 5x 10x 1 76. 81c 36c 4 77. 9x 4x 16 78. 4y 0y 5 79. 16x 7x 81 80. 49b 11b 64 Group I 81. 1a 8a 8 3 8. 4z 16z 15z 83. 1n 30n 1 3 84. 1t 54t 4t 69. 64y 5 85. 4 3 15x 5x 10x 51. 1x 0x 5 70. x 4 81 86. 8m 4mn 1n 5. 1y 19y 1 53. 0m mn 1n 54. 4t 11t 18 55. 0t t 30 56. 10x 9x 36 57. 1x 3xy 4y Group H 71. 9x 30x 5 7. 4n 8n 49 87. 30z 5z 0 88. 100x 10x 0 3 89. 1y 9y 30y 90. 48x 60x 18 1
Group J 91. 1x 18x 1 9. 90y 195y 100 93. 94. 3 48a 8a b 4ab 4x 36xy 1y 3 95. 0t 31t 1t 3 96. 4w 36w 4w 3 97. 36x 87x 45x 98. 48x 64x 40 3 99. 1x y x y 4xy 100. 7m 54m 180 3 13