1. UNIT 14: LESSON 1 Fundamentals of “box” factoring trinomials Sum & product practice

2. Factoring is the reverse process of multiplication. From a previous lesson, recall the “box” method that can be used to multiply these two binomials: (x + 2)(x – 5) x x 2 -5 x2x2 2x -5x-10 = x 2 – 5x + 2x – 10 = x 2 – 3x – 10

3. Take special note of two important details in the box: Notice the product of the items in each of the diagonals is the same. Notice the sum of the items in the positive slope diagonal is the middle term of the answer to (x + 2)(x – 5) = x 2 – 3x - 10 x x 2 -5 x2x2 2x -5x-10 (-5x)(2x) = -10x 2 (x 2 )(-10) = -10x 2 x x 2 -5 x2x2 2x -5x-10 -5x + 2x = -3x

4. Now think backwards on this process in order to factor x 2 – 3x -10. Place the first and last terms in the boxes opposite each other and write their product. x2x2 -10 x 2 – 3x – 10 (x 2 )(-10) = -10x 2

5. Next, fill in the other two diagonally opposite corners. The requirement for them is that their product equal the previous diagonal product and their sum be equal to the “middle term,” -3x. List numbers that multiply to -10 and check to see if they also add to -3. So, for our problem, the two question marks are 2x and -5x. Product ➤ (?)(?) = -10x 2 Sum ➤ ? + ? = -3x ? ? -2 x 5 = -10-2 + 3 = 1 2 x -5 = -102 + -5 = -3

6. Now that our box is complete, how does this help us factor the problem? Each question mark below represents the GCF of its corresponding row or column. For the factors of our original problem (x 2 – 3x + 10), just read x + 2 off of the top of the box and x – 5 off of the left of the box. x 2 – 3x -10 = (x + 2)(x – 5) ? ? ? ? x2x2 2x -5x-10 x x 2 -5 x2x2 2x -5x-10 x & 2 are the GCFs of their columns. x & -5 are the GCFs of their rows. factors

7. As demonstrated in the preceding steps, it is observed that one of the main tasks in the process of factoring is the finding of two quantities that produce both a specified sum and product. Practice on that task is the main focus of this lesson.