Lesson Factoring trinomials by the ac-method and grouping

Source code of 'Factoring trinomials by the ac-method and grouping'

This Lesson (Factoring trinomials by the ac-method and grouping) was created by by Edwin McCravy(20086)

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(A) Factor: 9x�-13x-10   (Or "factorISE" it if you live in the UK)

Multiply the 9 by the 10 ignoring signs.  Get 90

Write down all the ways to have two positive integers
which have product 90, starting with 90*1

90*1
45*2
30*3
18*5
15*6
10*9

Since the last sign in 9x�-13x-10 is -, SUBTRACT them,
and place the DIFFERENCE out beside that:

90*1   90-1=89
45*2   45-2=44
30*3   30-3=27
18*5   18-5=13
15*6   15-6= 9 
10*9   10-9= 1

Now, again ignoring signs, we find in that list of
differences the coefficient of the middle term in 9x�-13x-10

So we replace the number 13 by 18-5

9x�-13x-10
9x�-(18-5)x-10

Then we distribute to remove the parentheses:

9x�-18x+5x-10

Factor the first two terms 9x�-18 by taking out the
greatest common factor, 9x, getting 9x(x-2)

Factor the last two terms +5x-10 by taking out the
greatest common factor, getting +5(x-2)

So we have

9x(x-2)+5(x-2)

Notice that there is a common factor, (x-2)

9x(x-2)+5(x-2)

which we can factor out leaving the 9x and the +5 to put 
in parentheses:

(x-2)(9x+5)

---------------------------

(B) Factor:  12x�-32x+21

Multiply the 12 by the 21 ignoring signs.  Get 252

Write down all the ways to have two positive integers
which have product 252, starting with 252*1

252*1
126*2
 84*3
 63*4
 42*6
 36*7
 28*9
 21*12
 18*14

Since the last sign in 12x�-32x+21 is +, ADD them,
and place the SUM out beside that:

252*1    252+1=253
126*2    126+2=128
 84*3     84+3-87
 63*4     63+4=67
 42*6     42+6=48
 36*7     36+7=43
 28*9     28+9=37 
 21*12   21+12=33
 18*14   18+14=32

Now, again ignoring signs, we find in that list of
sums the coefficient of the middle term in 12x�-32x+21

So we replace the number 32 by 18+14

12x�-32x+21
12x�-(18+14)x+21

Then we distribute to remove the parentheses:

12x�-18x-14x+21

Factor the first two terms 12x�-18x by taking out the
greatest common factor, getting 6x(2x-3)

Factor the last two terms -14x+21 by taking out the
greatest common factor, -7, getting -7(2x-3)

So we have

6x(2x-3)-7(2x-3)

Notice that there is a common factor, (2x-3)

6x(2x-3)-7(2x-3)

which we can factor out leaving the 6x and the -7 to put 
in parentheses:

(2x-3)(6x-7)

Edwin</pre>