Preliminary Exercises for Factoring Quadratics (Two Numbers)

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1 #3160b GLA Guided Learning Activity Preliminary Exercises for Factoring Quadratics (Two Numbers) This packet contains exercises to accompany GLA It is intended for student and tutor use as needed. Author: Dennis Morrow SIGMA-MAC College of the Canyons Mathematics Supplement Preliminary Exercises for Factoring Quadratics Example 1: Find all sums and differences that can be made by multiplying factors of 6 by factors of 12. The factors of 6 could be 1 and 6 or 2 and 3, and the factors of 12 could be 1 and 12 or 2 and 6 or 3 and 4. Here is the methodical approach, starting with (1,6) beside all orderings of the factors of 12, and multiplying the outside pair of numbers together and the inside numbers together: Factorizations Outer and Inner Products Sums Differences (1,6)(12,1) 1 1 = 1, 6 12 = = = 71 (1,6)(1,12) 1 12 = 12, 6 1 = = = 6 (1,6)(6,2) 1 2 = 2, 6 6 = = = 34 (1,6)(2,6) 1 6 = 6, 6 2 = = = 6 (1,6)(4,3) 1 3 = 3, 6 4 = = = 21 (1,6)(3,4) 1 4 = 4, 6 3 = = = 14 (2,3)(12,1) 2 1 = 2, 3 12 = = = 34 (2,3)(1,12) 2 12 = 24, 3 1 = = = 21 (2,3)(6,2) 2 2 = 4, 3 6 = = = 14 (2,3)(2,6) 2 6 = 12, 3 2 = = = 6 (2,3)(4,3) 2 3 = 6, 3 4 = = = 6 (2,3)(3,4) 2 4 = 8, 3 3 = = = 1

2 So, the only possible sums of the products of the factors are 17, 18, 22, 27, 38, and 73. The only possible differences are 1, 6, 14, 21, 34, and 71. If asked for a sum or difference other than these numbers your reply will be NP for not possible. Notice that 18 occurs four times as a sum. Multiple answers are possible now, but they will not occur when we apply this technique to factor algebraic expressions later. Multiple answers occur when the two numbers to be factored and the proposed sum or difference have a common factor other than 1. Example 2: Find factors of 6 and factors of 12 which when cross multiplied have a sum of 73. If we do not see that the 6 should be 6 times 1 and the 12 should be 12 times 1 and that the 6 and 12and the 1 and 1 will be multiplied to form the sum of 72 and 1, then we start listing the possibilities as in the example above until we find the right combination. It is at the top of the list as (1,6)(12,1). However, as practice for things to come, this is not the order desired for the answer. Simply switch the middle two numbers. The answer is (1,12)(6,1). The (1,6)(12,1) form was very convenient for listing the factors and keeping track of the orderings, but the (1,12)(6,1) form is the way the application later will take. This form has the factors of the first number, 6, as the first component in each ordered pair and the factors of the second number, 12, as the second component in each ordered pair. The first numbers in each ordered pair multiply to give the first number given, i.e. 6. The outside numbers multiply to give 1 and the inside numbers multiply to give 72, the sum of which is 73. The last numbers multiply to give the second given number, i.e.12. The pattern is: First, Outside, Inside, Last, or FOIL. Example 3: Find factors of 6 and factors of 10 which when cross multiplied have a difference of 11. (2,5)(3,2) Checking: First 2 3 = 6 Outside 2 2 = 4, Inside 5 3 = 15, difference 11 Last 5 2 = 10 Answer is correct.

3 Exercises: If not possible answer ( N, P )(, ) 1. Find factors of 6 and factors of 10 which when cross multiplied have a sum of Find factors of 6 and factors of 10 which when cross multiplied have a difference of Find factors of 5 and factors of 10 which when cross multiplied have a sum of Find factors of 5 and factors of 10 which when cross multiplied have a difference of Find factors of 2 and factors of 7 which when cross multiplied have a sum of Find factors of 2 and factors of 7 which when cross multiplied have a difference of Find factors of 3 and factors of 10 which when cross multiplied have a sum of Find factors of 3 and factors of 10 which when cross multiplied have a difference of Find factors of 4 and factors of 5 which when cross multiplied have a sum of Find factors of 4 and factors of 5 which when cross multiplied have a difference of 2.

4 11. Find factors of 6 and factors of 18 which when cross multiplied have a sum of Find factors of 6 and factors of 18 which when cross multiplied have a difference of Find factors of 6 and factors of 15 which when cross multiplied have a sum of Find factors of 6 and factors of 10 which when cross multiplied have a difference of Find factors of 12 and factors of 12 which when cross multiplied have a sum of Find factors of 12 and factors of 12 which when cross multiplied have a difference of Find factors of 5 and factors of 36 which when cross multiplied have a sum of Find factors of 5 and factors of 36 which when cross multiplied have a difference of Find factors of 14 and factors of 10 which when cross multiplied have a sum of Find factors of 14 and factors of 10 which when cross multiplied have a difference of 11.

5 21. Find factors of 14 and factors of 15 which when cross multiplied have a sum of Find factors of 14 and factors of 15 which when cross multiplied have a difference of Find factors of 16 and factors of 9 which when cross multiplied have a sum of Find factors of 16 and factors of 9 which when cross multiplied have a difference of Find factors of 4 and factors of 25 which when cross multiplied have a sum of Find factors of 4 and factors of 25 which when cross multiplied have a difference of Find factors of 6 and factors of 35 which when cross multiplied have a sum of Find factors of 6 and factors of 35 which when cross multiplied have a difference of Find factors of 16 and factors of 81 which when cross multiplied have a sum of Find factors of 16 and factors of 81 which when cross multiplied have a difference of 54.