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A rational expression
is a ratio of polynomials:
Examples:
Whenever an expression containing variables is present in the denominator of
a fraction, you should be alert to the possibility that certain values of the
variables might make the denominator equal to zero, which is forbidden. This
means that when we are talking about rational expressions we can no longer say
that the variable represents “any real number.” Certain values may have to be
excluded. For example, in the expression
 ,
we cannot allow the value x = 0 so we would parenthetically
add the comment (x ¹ 0), and for
we would say (x ¹ 3) . In the case of
we would exclude both x = 1 and x = –1,
since either choice would make the denominator zero.
We don’t care if the numerator is zero. If the numerator is zero,
that just makes the whole rational expression zero (assuming, of course, that
the denominator is not zero), just as with common fractions. Recall that
0/4 = 0, but 4/0 is undefined.
It is important to keep this in mind as you work with rational expressions,
because it can happen that you are trying to solve an equation and you get one
of the “forbidden” values as a solution. You would have to discard that
solution as being unacceptable. You can also get some crazy results if you
don’t pay attention to the possibility that the denominator might be zero for
certain values of the variable. For example, the celebrated proof that
0 = 1 and other nonsense.
Proof that 1 = 0
And other nonsense
Can you identify the
flaw in this argument?
Let x = 1. Then
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Given:
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x = 1
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Multiply both sides by x:
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x2 = x
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Subtract x from both sides:
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x2 – x = 0
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Factor out an x:
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x(x – 1) = 0
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Divide both sides by (x ‑ 1):
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x = 0
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But x = 1, so substitute 1 for x to get:
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1 = 0
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This is a very simple variant of this classic
‘proof’. Once you see the trick*, you can construct more elaborate versions
that do a better job of concealing the error, and you can vary it to ‘prove’
other nonsense such as 1 = 2.
For example:
Let x = 1 . Then
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Given:
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x = 1
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Multiply both sides by –1:
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-x = –1
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Add x2 to both sides:
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x2 - x = x2 – 1
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Factor both sides:
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x(x – 1) = (x – 1)(x + 1)
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Divide both sides by (x – 1):
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x = (x + 1)
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Substituting 1 back in for x gives the result:
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1 = 2
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*The trick:
We are dividing by zero because if x = 1 then (x – 1) = 0.
Thus, all of these “proofs” are invalid because they use an illegal step.
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