The Australian Method
It involves some good old fashioned logic, as well as some ideas from the method of decomposition.
We will work thru an example, step by step, then we will throw one in, maybe two......full speed.
I will make notes at each step, for your understanding.
Factor 6 x2  29 x  35. We know it will be the product of two binomials.
6 x2  29 x  35  (

)...now just use the first coefficient along with x , in each bracket.
)(
(6 x
)(6 x
6
)
......in the brackets, (6 x)(6 x)  36 x 2 ...which is too much so ÷by 6.
Now we use decomposition: first times last, 6×35 = 210 and the middle coefficient, 29.
Now use your factoring skills to determine two numbers with a product of 210 and a sum of 29.
They are +14 and +15. So insert them, in any order, into the brackets.
 6 x 2  29 x  35 
(6 x  14)(6 x  15)
.......now what divides into both 6x and 14. It is 2.
6
Now use that 2 to express 6 as a product of two factors.....6 = 2×3 and rewrite as follows....
 6 x 2  29 x  35 
(6 x  14)(6 x  15)
......now divide 2 into the first binomial and 3 into the second.
(2)(3)
 6 x2  29 x  35  (3x  7)(2 x  5) .....same result as the two previous methods.
This method only takes a few lines....a lot of the work is mental arithmetic.
Some food for thought.....there is no “best” method. It’s whatever works for you. It’s a good idea to
explore different methods and decide which you prefer. Some methods work better under certain
circumstances.
We will move on to another example but I want to point out another detail. It is more difficult to explain
these techniques on paper on a Wiki, than it is to explain them in front of a class. It is important to be
able to ask questions and have class interaction.
In any case, I thought that it is important to offer lessons on many topics, in a variety of ways.
I hope that you enjoy reading and learning.......
Now, a little more practice.
Factor 14 x 2  13x  12. Again I will try to include a few steps......14×(-12) = -168.
We need two numbers that multiply to -168 and add to -13. I will let you list the possibilities in the
search: the numbers are 8 and -21. The order is not important....I will let you discover that for yourself.
14 x 2  13x  12 
(14 x  21)(14 x  8)
.......now we will use 14 = 7×2
14
14 x 2  13x  12 
(14 x  21)(14 x  8)
(7)(2)
14 x2  13x  12  (2 x  3)(7 x  4)
One last one....which is the best method? Decomposition, Chart or Australian?
Factor 44 x 2  67 x  72. Let’s try all three methods.
Decomposition: 44×(-72) = -3168. Now what two numbers have a product of -3168 and a sum of -67.
This is what takes the longest time...for me anyway. After some “donkey work” we have 32 and -99.
 44 x 2  67 x  72  44 x 2  32 x  99 x  72
 4 x(11x  8)  9(11x  8)
 (11x  8)(4 x  9)
Once you have the two numbers......it is quite simple but it can take a lot of time to find them.
That is why a systematic approach is important.
Now let’s try Chart : Factor 44 x2  67 x  72
1
2
4
44
22 11
1
72
2
36
3
24
4
18
6
12
8
9
11×9 = 99
72
1
36
2
24
3
18
4
12
6
9
8
4×8 = 32
-99 +32 = -67  44 x2  67 x  72  (4 x  9)(11x  8)
This is the method that I would use since the product is large.
Now let’s try Australian: Factor 44 x2  67 x  72
44×(-72) = -3168......the two numbers that multiply to -3168 and add to -67 are 32 and -99 from the
page above, using Decomposition.
 44 x 2  67 x  72 
(44 x  32)(44 x  99)
44

(44 x  32)(44 x  99)
(4)(11)
 (11x  8)(4 x  9)
There is one more type that I want to show you in the next lesson.