Decomposition Method
This is essentially the same process as Sum Product Factoring which works as such:
- Identify \(a\), \(b\) and \(c\)
- Determine 2 integers whose Product is \(ac\) and Sum is \(b\). This may require some trial and error
- Once these values are found, split \(b\) into 2 terms based on these values
- Split the expression into 2 pairs. Factor the GCF out of both pairs
- Place the GCF's into another pair to get the fully factored expression
Example
Factor the expression \(2x^2 + 7x + 3\)
First, we can identify \(a = 2\), \(b = 7\) and \(c = 3\).
We need to determine 2 integers whose product is \((2)(3) = 6\) and sum is \(7\). We can identify these values as \(6\) and \(1\).
Now that we have our integers, we can split \(b\) into the respective terms:
\(2x^2 + 6x + x + 3\)
We can split the terms into 2 pairs and factor the GCF out of both to get the fully factored expression:
\((2x^2 + 6x) + (x + 3)\)
\(2x(x + 3) + 1(x + 3)\)
\((x + 3)(2x + 1)\)
Therefore, we can determine that \(2x^2 + 7x + 3\) fully factored is \((x + 3)(2x + 1)\).
Factor \(10x^4 - 3x^2 - 18\) using the Decomposition method.
Show Answer
First, we can identify \(a = 10\), \(b = -3\), and \(c = -18\).
We need to determine 2 integers whose product is \((10)(-18) = -180\) and sum is \(-3\). We can identify these values as \(-15\) and \(12\).
Now that we have our integers, we can split \(b\) into the respective terms:
\(10x^4 + 12x^2 - 15x^2 - 18\)
We can split the terms into 2 pairs and factor the GCF out of both to get the fully factored expression:
\((10x^4 + 12x^2) + (-15x^2 - 18)\)
\(2x^2(5x^2 + 6) - 3(5x^2 + 6)\)
\((5x^2 + 6)(2x^2 - 3)\)
Therefore, we can determine that \(10x^4 - 3x^2 - 18\) fully factored is \((5x^2 + 6)(2x^2 - 3)\).
Criss-Cross Method
This is another method of factoring which works using the following process:
- Identify \(a\), \(b\), and \(c\)
- Create a table of values to go through all possible integer combinations
- Once these integers are found, factor the expression by grouping
Example
Factor the expression \(6x^2 + 10x - 4\).
First, we can identify \(a = 6\), \(b = 10\) and \(c = -4\).
Therefore, we need to determine 2 integers whose product is \((6)(-4) = -24\) and sum is \(10\).
We can create a table to go through all possible combinations:
| r value |
1 |
-1 |
2 |
-2 |
3 |
-3 |
4 |
-4 |
| s value
| -24 |
24 |
-12 |
12 |
-8 |
8 |
-6 |
6 |
| Sum
| -23 |
23 |
-10 |
10 |
-5 |
5 |
-2 |
2 |
| Product
| -24 |
-24 |
-24 |
-24 |
-24 |
-24 |
-24 |
-24 |
Using the table, we can identify the 2 integer values as \(-2\) and \(12\). We can rewrite the expression as such to fully factor:
\(6x^2 + 12x - 2x - 4\)
\((6x^2 + 12x) + (-2x - 4)\)
\(6x(x + 2) - 2(x + 2)\)
\((x + 2)(6x - 2)\)
Therefore, using the Criss-Cross Method, we can determine that \(6x^2 + 10x - 4\) fully factored is \((x + 2)(6x - 2)\).
Factor \(12r^2 + 7rs - 10s^2\) using the Criss-Cross Method.
Show Answer
First, we can identify \(a = 12\), \(b = 7\) and \(c = -10\).
Therefore, we need to determine 2 integers whose product is \((12)(-10) = -120\) and sum is \(7\).
We can create a table to go through all possible combinations:
| r value |
1 |
-1 |
2 |
-2 |
3 |
-3 |
4 |
-4 |
5 |
-5 |
6 |
-6 |
8 |
-8 |
10 |
-10 |
| s value
| -120 |
120 |
-60 |
60 |
-40 |
40 |
-30 |
30 |
-24 |
24 |
-20 |
20 |
-15 |
15 |
-12 |
12 |
| Sum
| -119 |
119 |
-58 |
58 |
-37 |
37 |
-26 |
26 |
-19 |
19 |
-14 |
14 |
-7 |
7 |
-2 |
2 |
| Product
| -120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
-120 |
Using the table, we can identify the 2 integer values as \(-8\) and \(15\). We can rewrite the expression as such to fully factor:
\(= 12x^2 + 15x -8x - 10\)
\(= (12x^2 + 15x) + (-8x - 10)\)
\(= 3x(4x + 5) - 2(4x + 5)\)
\(= (4x + 5)(3x - 2)\)
Therefore, using the Criss-Cross Method, we can determine that \(12r^2 + 7rs - 10s^2\) fully factored is \((4x + 5)(3x - 2)\).
Australian Method
This is the last method of factoring which works using the following process:
- Identify \(a\), \(b\), and \(c\)
- Divide the entire expression by \(a\)
- Determine 2 integer values whose product is \(ac\) and sum is \(b\). This may require some trial and error.
- Create a factored expression in the form \((x + r)(x + s)\) as the numerator
- Factor out the GCF of both pairs of binomials
- Divide the GCF's of by the denominator to get the fully factored expression
Example
Factor the expression \(56x^2 - 9x - 2\)
First, we can identify \(a = 56\), \(b = -9\), and \(c = -2\).
Next, we need to divide the entire expression by \(56\):
\(\cfrac{56x^2 - 9x - 2}{56}\)
Then, we need to determine 2 integer values whose product is \(-112\) and sum is \(-9\).
We can identify \(r\) and \(s\) as \(7\) and \(-16\) respectively.
Using these values, we can rewrite the numerator as such:
\(\cfrac{(56x + 7)(56x - 16)}{56}\)
We can factor the GFC's out of the respective sets of binomials then divide those GFC's by the denominator:
\(= \cfrac{7(8x + 1)8(7x - 2)}{56}\)
\(= (8x + 1)(7x - 2)\)
Therefore, we can determine that \(56x^2 - 9x - 2\) fully factored is \((8x^2 + 1)(7^2 - 2)\).
Factor \(20x^6 - 59x^3y^2 + 42y^4\) using the Australian method
Show Answer
First, we can identify \(a = 20\), \(b = -59\), and \(c = 42\).
Next, we need to divide the entire expression by 56:
\(\cfrac{20x^6 - 59x^3y^2 + 42y^4}{20}\)
Then, we need to determine 2 integer values whose product is (20)(42) = 840 and sum is -59. We can identify these values as -35 and -24.
Using these values, we can rewrite the numerator as such:
\(\cfrac{(20x^3 - 35y^2)(20x^3 - 24y^2)}{20}\)
We can factor the GFC's out of the respective sets of binomials then divide those GFC's by the denominator:
\(= \cfrac{5(4x^3 - 7y^2)4(5x^3 - 6y^2)}{20}\)
\(= (4x^3 - 7y^2)(5x^3 - 6y^2)\)
Therefore, we can determine that \(20x^6 - 59x^3y^2 + 42y^4\) fully factored is \((4x^3 - 7y^2)(5x^3 - 6y^2)\).
The height, \(h\), in metres, of a baseball above the ground relative to the horizontal distance, \(d\), in metres, from the batter is given by \(h = -0.005d^2 + 0.49d + 1\)
i. Write the right side of the equation in factored form.
ii. At what horizontal distance from the batter will the baseball hit the ground if its not caught by an outfielder?
Show Answer
i. First, we can start factoring the expression by factoring out -0.005
\(h = -0.005(d^2 - 98d - 200)\)
Now, we can identify \(a = 1\), \(b = -98\) and \(c = -200\).
As a result, we need to determine 2 integer values whose product is (1)(-200) = -200 and sum is -98.
We can create a table to go through all possible combinations:
| r value |
1 |
-1 |
2 |
-2 |
4 |
-4 |
5 |
-5 |
8 |
-8 |
10 |
-10 |
| s value
| -200 |
200 |
-100 |
100 |
-50 |
50 |
-40 |
40 |
-25 |
25 |
-20 |
20 |
| Sum
| -199 |
199 |
-98 |
98 |
-46 |
46 |
-35 |
35 |
-17 |
17 |
-10 |
10 |
| Product
| -200 |
-200 |
-200 |
-200 |
-200 |
-200 |
-200 |
-200 |
-200 |
-200 |
-200 |
-200 |
Using the table, we can identify the 2 integer values as 2 and -100. We can rewrite the expression as such to fully factor:
\(h = -0.005(d^2 + 2d - 100d - 200)\)
\(h = -0.005((d^2 + 2d) + (-100d - 200))\)
\(h = -0.005(d(d + 2) -100(d + 2))\)
\(h = -0.005((d + 2)(d - 100))\)
Therefore, we can determine that \(h = -0.005d^2 + 0.49d + 1\) fully factored is \(h = -0.005((d + 2)(d - 100))\).
ii. Using the factored expression, we can determine that the factors (or zeroes) are \(-2\) and \(100\).
As distance cannot be negative, we can identify that if the baseball will reach a horizontal distance of \(100\;[m]\) if it isn't caught by an outfielder.
This can be illustrated using a graph:
The area of a rectangular parking lot is represented by the expression \(6x^2 - 19x - 7\)
i. Factor the expressions for length and width
ii. If \(x\) represents \(15\;[m]\), what are the length and width of the parking lot?
Show Answer
i. First, we can identify \(a = 6\), \(b = -19\), and \(c = -7\).
As a result, we need to determine 2 integer values whose product is \((6)(-7) = -42\) and sum is \(-19\).
We can create a table to go through all possible combinations:
| r value |
1 |
-1 |
2 |
-2 |
3 |
-3 |
6 |
-6 |
| s value
| -42 |
42 |
-21 |
21 |
-14 |
14 |
-7 |
7 |
| Sum
| -41 |
41 |
-19 |
19 |
-11 |
11 |
-1 |
1 |
| Product
| -42 |
-42 |
-42 |
-42 |
-42 |
-42 |
-42 |
-42 |
Using the table, we can identify the 2 integer values as \(2\) and \(-21\). We can rewrite the expression as such to fully factor:
\(A = 6x^2 + 2x - 21x - 7\)
\(A = (6x^2 + 2x) + (-21x - 7)\)
\(A = 2x(3x + 1) -7(3x + 1)\)
\(A = (3x + 1)(2x - 7)\)
Therefore, we can determine that \(A = 6x^2 - 19x - 7\) fully factored is \(A = (3x + 1)(2x - 7)\).
ii. For this question, all we need to do is plug \(15\;[m]\) into both factors to find the respective side lengths:
\(\text{Length} = 3x + 1\)
\(\text{Length} = 3(15) + 1\)
\(\text{Length} = 45 + 1 = 46\)
\(\text{Width} = 2x - 7\)
\(\text{Width} = 2(15) - 7\)
\(\text{Width} = 30 - 7 = 23\)
Therfore, we can determine that the length and width are \(46\;[m]\) and \(23\;[m]\) respectively.