Factoring using the box method is probably the best way to factor a trinomial of the form
ax^{2} + bx + c.
The box is just a 2x2 square that we use to put the terms of the trinomial.
Example #1
Factor 2x^{2} + 9x + 10 using the box method.
Step 1
First, put 2x^{2} and 10 in the box below as shown. You will always put the first term in the first row and the first column and the last term in the second row and second column.
Step 2
Multiply the first term by the last term: 2x^{2 }×10 = 20x^{2}
Look for factors of 20x^{2} that will add up to the second term or 9x.
20x^{2} = 1x × 20x
20x^{2} = 2x × 10x
20x^{2} = 4x × 5x
Since 4x and 5x add up to 9x, put 4x and 5x in the box.
Step 3
Find the greatest common factor of each row and write them down on the left or right side of the box. Then, find the greatest common factor of each column and write them down on top or at the bottom of the box.
Notice the following:
2x times x = 2x^{2}
2x times 2 = 4x
5 times x = 5x
5 times 2 = 10
The factors are (2x + 5) and (x + 2), so 2x^{2} + 9x + 10 = (2x + 5) × (x + 2)
4x and 5x are interchangeable, so you could put 5x in the first row and 4x in the second row and still get the same answer.
When factoring using the box method, you need to make sure that the greatest common factor of a, b, and c in ax^{2} + bx + c is equal to 1 as example #2 shows.
Example #2
Factor 6x^{2} + 27x + 30 using the box method.
Step 1
First put 6x^{2} and 30 in the box below as
shown.
Step 2
Multiply the first term by the last term: 6x^{2 }× 30 = 180x^{2}
Look for factors of 180x^{2} that will add up to 27x.
Since 12x × 15x = 180x^{2} and 12x + 15x = 27x, put 12x and 15x in the box.
Step 3
Find the
greatest common factor of each row and write them down on the left or
right side of the box. Then, find the greatest common factor of each
column and write them down on top or at the bottom of the box. We end up with the following box.
However, we have a little problem . 6x times 3x is not equal to 6x^{2}, 6x times 6 is not equal to 12x, 15 times 3x is not equal to 15x, and 15 times 6 is not equal to 30.
In order for us to see what happened here, we need to make an important observation.
Notice that 6x^{2} + 27x + 30 = 3(2x^{2} + 9x + 10 ) and we already factored 2x^{2} + 9x + 10
2x^{2} + 9x + 10 = (2x + 5)× (x + 2)
Therefore, 6x^{2} + 27x + 30 = 3(2x^{2} + 9x + 10 ) = 3(2x + 5)×(x + 2)
Looking at the box immediately above again, we see that we can factor the entire thing by 3. And if we do that we will get the same answer as illustrated below.
Look at the image immediately above again. The box inside the parentheses is the box that we used to factor 2x^{2} + 9x + 10.
Therefore when factoring using the box method, make sure you factor the trinomial ax^{2} + bx + c until the greatest common factor of a, b, and c is equal to 1 to avoid complicating things.
Example #3
Factor 4x^{2} - 8x - 12 using the box method.
First, factor 4x^{2} - 8x - 12 using the greatest common factor.
4x^{2} - 8x - 12 = 4(x^{2} - 2x - 3)
Now, you just need to factor x^{2} - 2x - 3 using the box method
Step 1
First put x^{2} and -3 in the box
Step 2
Multiply the first term by the last term: x^{2 }× -3 = -3x^{2}
Look for factors of -3x^{2} that will add up to -2x.
Since -3x × x = -3x^{2} and -3x + x = -2x put -3x and x in the box.
Step 3
Find the
greatest common factor of each row and write them down on the left or
right side of the box. Then, find the greatest common factor of each
column and write them down on top or at the bottom of the box. We end up with the following box.
x^{2} - 2x - 3 = (x + 1)(x - 3)
4x^{2} - 8x - 12 = 4(x^{2} - 2x - 3) = 4(x + 1)(x - 3)
Apr 02, 20 12:05 PM
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