SOLUTION: In factoring using the difference of cubes formula, if you have the problem 1+8x^3 do you need to switch it to 8x^3+1 to work or can you solve it the way it is written?

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Question 1190174: In factoring using the difference of cubes formula, if you have the problem 1+8x^3 do you need to switch it to 8x^3+1 to work or can you solve it the way it is written?
Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Either way works perfectly fine. This is because adding any two numbers can be done in any order you want.
A+B = B+A
Furthermore, 8x^3+1 will factor the same way as 1+8x^3

I should point out however, that the 8x^3+1 is a sum of cubes (not a difference of cubes).

The sum of cubes factoring formula is
a^3 + b^3 = (a+b)(a^2 - ab + b^2)

One example of using that formula
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
(4x)^3 + (2)^3 = (4x+2)((4x)^2 - (4x)*2 + 2^2)
64x^3 + 8 = (4x+2)(16x^2 - 8x + 4)

Notice how swapping the roles of 'a' and b doesn't matter
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
2^3 + (4x)^3 = (2+4x)(2^2 - 2*4x + (4x)^2)
8 + 64x^3 = (2+4x)(4 - 8x + 16x^2)
8 + 64x^3 = (4x+2)(16x^2 - 8x + 4)
Showing we get the same exact factorization as earlier.

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A related factoring formula is the difference of cubes factoring formula
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
In this case, order does matter because a^3-b^3 is not the same as b^3-a^3 (much like how 10-3 = 7 is not the same as 3-10 = -7)

An example:
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
(7x)^3 - 5^3 = (7x-5)((7x)^2 + 7x*5 + 5^2)
343x^3 - 125 = (7x-5)(49x^2 + 35x + 25)

If we swapped 'a' and b, then,
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
(5)^3 - (7x)^3 = (5-7x)((5)^2 + 5*7x + (7x)^2)
125 - 343x^3 = (5-7x)(25 + 35x + 49x^2)
125 - 343x^3 = (5-7x)(49x^2 + 35x + 25)
It's very close to the earlier factorization in the previous paragraph.
However, the key difference is that the 7x-5 is not the same as 5-7x.
Therefore, the two factorizations aren't the same.
Again it's all because the order of subtraction matters.
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

In factoring using the difference of cubes formula, if you have the problem 1+8x^3 do you need to switch it to 8x^3+1 to work or can you solve it the way it is written?
Given in that form (variable, preceded by the constant), it should be left as is. If it's a multiple-choice question, the 

answer would most likely be in that form also. Plus, you apply the same concept to get your answer. Below you'll find both forms: 

 

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