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into each column (see Table 1).

clude the middle term. The sum of the 2nd row and the sum of the 3rd row are multiplied together to produce the given trinomial. That is, (2 x – 1) multi plied by (3 x + 4) = (6 x ) 2 + 5 x – 4. Step 7: If the sum is not equal to the middle term of the polynomial, try the other possible factor combi nations for the leading term and the constant term, using a new Tic Tac Toe table. Repeat this process until you find the sum of the second column that is equal to the middle term of the polynomial. Then, go back to Step 6.

Table 1

Step 2: Factor the leading term and the constant term. The leading term factors are 6 x 2 = 2 x × 3 x or x × 6 x and the constant term factors are - 4 = 1 × - 4, - 4 × 1, - 1 × 4, 4 × - 1, - 2 × 2, or 2 × - 2. Step 3: Fill the Tic Tac Toe table row in column one and column three with one possible solution by using one factor from the leading term and one fac tor from the constant term. For example, we have 2x and 3x for the leading term in column one in rows two and three. We have the constant terms, - 1 and 4, in column 3 in rows two and three (see Ta ble 2).

For example, we use one possible pair, in which the leading term factors 2 x and 3 x are paired with the constant term factors - 1 and 4. Place the lead ing term factors under the first column in rows two and three and the constant factors under the third column in rows two and three in the Tic Tac Toe table. In the center column, we place the product of the cross - multiplied values, (2x)(4) = 8x, which is placed in row 2 in column 2 and (3x)( - 1) = - 3x is place in row 3 in column 2. In the third row under the second column we place the sum of the two terms above, 8x + ( - 3x) = 5x. Since 5 matches the value b in our trinomial, the factors are (2x + 4) and (3x - 1). Other possible Tic Tac Toe tables, although incorrect, are shown below.

Table 2

Step 4: Cross multiply one of the learning term fac tors with one of the last term factors and place the results into the middle column on the Tic Tac Toe table. Note that the multiplied terms from the dif ferent columns cannot reside on the same row. For example, 2 x × 4 = 8 x , which we place in row 2 under column 2. For 3 x × - 1 = - 3 x , it is placed in row 3 under column 2. Then, the sum of the prod ucts is 8 x + ( - 3 x ) = 5 x and placed in row 4 under column 2 (see Table 3).

Table 3

Step 5: Check to ensure the sum of the products of the cross - multiplications is equal to the middle term, b, of the polynomial. If so, these will be the factors for the trinomial. In this case, the sum is 5 x , which indicates they are the correct factors. If the cross multiplication does not produce the addends for the middle term, then go to Step 7.

Step 6: Write the sum of factors in the 2nd row and 3rd row in two separate parentheses. Do not in

Virginia Mathematics Teacher vol. 46, no. 2

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