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This step is not the source of the fallacy.This follows from
the previous step (2 a^2 - 2 ab = a^2 - ab) by factoring out
the 2 from the left-hand side and factoring out a 1 from the
The distributive law says that x(y+z) = xy + xz for all numbers
x, y, and z, and the factorization 2 a^2 - 2 ab = 2 (a^2 - ab)
is simply an application of this to the particular case x=2, y=a^2,
and z = -ab.
Writing a^2 - ab = 1(a^2 - ab) is legitimate since multiplying any
number by 1 leaves the number unchanged.
Why don't you go back to the list of steps in the
proof and see if you can identify which one
is wrong, now that you know it isn't this one?
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