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Free Homework Math Help: Factoring By The Table Method Or ACDD

 

Table Method Or ACDD

The next method of factoring is the table method or ACDD method. This refers to the general form of the trinomial:

               ax2 + bx + c

The AC part comes from multiplying the a and the c coefficients and the DD part comes from splitting the middle term in the general form to produce a new form:

               ax2 + d1x + d2x + c

               d1 + d2 = b

The idea at this point is to factor the product ac into pairs of factors. These factors become d1 and d2 and are added together and compared to the middle coefficient b. If the sum of d1 and d2 are found then d1 and d2 become the coefficients of the new form. The rest of the problem is just factoring by grouping. A table is often used to keep track of the pairs of factors and their sums. Example:

               28x2 + 79x + 55

               a = 28                b = 79                c = 55

               ac = 28 * 55 = 1540

Create a table to keep track of factors and their sums.

Note that ac is the same as d1d2, Also, d1 and d2 are the pairs of factors. When listing the factors, start with the first factor equal to one and increment. In addition, it can help to factor a first then c and list the factors from this combined set.

a = 28 = 1*2*2*7 ac = d1d2
c = 55 = 1*5*11 28 * 55 = 1540       b = 79
ac = 28*55 = 1*2*2*5*7*11         d1*d2 d1 + d2 = b
1*2*2*5*7*11 1*1540 1 + 1540 = 1541
1*2*2*5*7*11 2*770 2 + 770 = 772
1*2*2*5*7*11 4*385 4 + 385 = 389
1*2*2*5*7*11 5*308 5 + 308 = 313
1*2*2*5*7*11 7*220 7 + 220 = 227
1*2*2*5*7*11 10*154 10 + 154 = 164
1*2*2*5*7*11 11*140 11 + 140 = 151
1*2*2*5*7*11 20*77 20 + 77 = 97
1*2*2*5*7*11 28*55 28 + 55 = 83
1*2*2*5*7*11 35*44 35 + 44 = 79 correct
1*2*2*5*7*11 55*28 55 + 28 = 83
1*2*2*5*7*11 77*20 70 + 20 = 97
1*2*2*5*7*11 140*11 140 + 11 = 151
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When the factors start to repeat it is possible to stop if their sum adds to the value of b. For negative factors, they can switch places without the minus sign moving to get the proper sign in the result.

When factors d1 and d2 are found, they are substituted into the modified general equation:

               ax2 + d1x + d2x + c

From the example above:

               a  = 28                d1 = 35                d2 = 44                c  = 55

The result of substitution is:

               28x2 + 35x + 44x + 55

Next, the first two terms are grouped together then the last two terms are grouped together:

               (28x2 + 35x) + (44x + 55)

The common factors in each group are factored out:

               7x(4x + 5) + 11(4x + 5)

The resulting two terms also have a common factor which is (4x + 5). This is factored out leaving the final answer in factored form:

               (7x + 11)(4x + 5)