Trinomial Factoring
Trinomials are polynomial expressions that contain 3 terms. Outlined below are the different methods used to factor trinomials.
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 \(c = 3\).
Next, we need to determine \(2\) integers whose product is \((2)(3) = 6\) and sum is \(7\). We can identify these values as \(6\) and \(1\).
Then, we can split \(b\) into the respective terms now that we have our integers:
Finally, 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 \(\boldsymbol{(x + 3)(2x + 1)}\).
First, we can identify \(a = 10\), \(b = -3\), and \(c = -18\).
Next, we need to determine \(2\) integers whose product is \((10)(-18) = -180\) and sum is \(-3\). We can identify these values as \(-15\) and \(12\).
Then, we can split \(b\) into the respective terms now that we have our integers:
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 \(\boldsymbol{(5x^2 + 6)(2x^2 - 3)}\).
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 \(c = -4\).
Next, 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 \(\boldsymbol{(x + 2)(6x - 2)}\).
First, we can identify \(a = 12\), \(b = 7\) and \(c = -10\).
Next, 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 \(\boldsymbol{(4x + 5)(3x - 2)}\).
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 \(56\):
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:
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 \(\boldsymbol{(8x^2 + 1)(7^2 - 2)}\).
First, we can identify \(a = 20\), \(b = -59\), and \(c = 42\).
Next, we can divide the entire expression by \(20\):
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:
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 \(\boldsymbol{(4x^3 - 7y^2)(5x^3 - 6y^2)}\).
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 \(h = -0.005d^2 + 0.49d + 1\)
- Write the right side of the equation in factored form
- At what horizontal distance from the batter will the baseball hit the ground if its not caught by an outfielder?
i. First, we can start factoring the expression by factoring out -0.005
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 \(\boldsymbol{h = -0.005((d + 2)(d - 100))}\).
ii. Using the factored expression, we can determine that the factors (or zeroes) are \(-2\) and \(100\).
As distance cannot be negative, we can determine that the baseball will reach a horizontal distance of \(\boldsymbol{100\;[\text{m}]}\) if it isn't caught by an outfielder.
This can be illustrated by sketching a graph:
The area of a rectangular parking lot is represented by the expression \(6x^2 - 19x - 7\)
- Factor the expressions for length and width
- If \(x\) represents \(15\;[\text{m}]\), what are the length and width of the parking lot?
i. First, we can identify \(a = 6\), \(b = -19\), and \(c = -7\).
Next, 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 \(\boldsymbol{A = (3x + 1)(2x - 7)}\).
ii. For this question, all we need to do is plug \(15\;[\text{m}]\) into both factors to find the respective side lengths.
We can start by determining the length:
\(l = 3x + 1\)
\(l = 3(15) + 1\)
\(l = 45 + 1\)
\(l = 46 \; [\text{m}]\)
Next, we can determine the width:
\(w = 2x - 7\)
\(w = 2(15) - 7\)
\(w = 30 - 7 = 23\)
\(w = 23\;[\text{m}]\)
Therfore, we can determine that the length and width are \(\boldsymbol{46\;[\text{m}]}\) and \(\boldsymbol{23\;[\text{m}]}\) respectively.