Use a proof by contradiction to show that there is no rational number r for which r+r+1=0. (Assume that r=a/h is a root, where a and b are integers and a/b is in lowest terms.obtain an equation involving integers by multiplying by then look at whether a and bare each odd or even.
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Use a proof by contradiction to show that there is no rational number for which
Proof
Assume that is a root, where and are integers and is in lowest terms.
We see that
if is odd, then is even,
if is even, then is odd.
Substitute
Multiply by
If is odd, is even, then
is odd, is even, is even. Hence the sum is odd. But zero is an even number.
We have a contradiction in this case.
If is even, is odd, then
is even, is even, is odd. Hence the sum is odd. But zero is an even number.
We have a contradiction in this case.
Therefore our assumption leads to the contradiction.
Therefore there is no rational number for which