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vortoca 2022-07-07 Answered
How does 1 x 2 + 1 x 3 become 1 2 x 1 3 x
I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where 1 x 2 + 1 x 3 becomes 1 2 x 1 3 x
The equations are obviously equal, but some algebraic manipulation is done between the first step and the second step, and I can't figure out what this manipulation could be.
The full breakdown comes from this solution
1 x 2 5 x + 6 = 1 ( x 2 ) ( x 3 ) = 1 3 ( 2 ) ( 1 x 2 1 x 3 ) = 1 x 2 + 1 x 3 = 1 2 x 1 3 x = n = 0 1 2 n + 1 x n n = 0 1 3 n + 1 x n = n = 0 ( 1 2 n + 1 1 3 n + 1 ) x n
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Answers (2)

Aryanna Caldwell
Answered 2022-07-08 Author has 11 answers
Each of the terms was multiplied by 1 1 , which is really equal to 1, so it's a "legal" thing to do:
1 x 2 + 1 x 3
= ( 1 ) 1 ( 1 ) ( x 2 ) + ( 1 ) 1 ( 1 ) ( x 3 )
= 1 2 x + 1 3 x
= 1 2 x 1 3 x
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Frederick Kramer
Answered 2022-07-09 Author has 7 answers
I am a grade 8 student, so I may not be able to explain really well.
First, I need to prove that 1 x 2 = 1 2 x
To prove, let's assume that "x" can be any number, for instance, I take x=8.
So by substituting,
1 x 2 = 1 8 2 = 1 6
And same for this,
1 2 8 = 1 6 = 1 6
Therefore, we have proven that 1 x 2 = 1 2 x
I also need to prove that 1 x 3 = 1 3 x
So by substituting,
1 8 3 = 1 5
and the same for this,
1 3 8 = 1 5 = 1 5 = 1 5
Therefore, we have proven that 1 x 3 = 1 3 x
By why it worked? The truth is, it is just having -1÷(-1)=1 (negative×negative=positive)(And anything times 1 is the same number)
So, from 1 x 2 to 1 2 x , they inserted both -1 for numerator and denominator as the following below.
1 x 2 = 1 x 2 = 1 ( 1 ) 1 ( x 2 ) = 1 x + 2 = 1 2 x
same goes to 1 x 3 = 1 3 x
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