Is the following an identity? Prove or disprove your claim:
This is a trivial question in junior form mathematics.
Many students may expand and re-group the polynomials so as to draw their conclusion, like
By their eyes, it is easy to draw the conclusion that
is NOT an identity since and are not identical.
Then, normally, I will add the comment:
for disproving identity, all we need to do is putting value(s) of x
and hope to see that
For example, put
since , is NOT an identity for sure.
Then a student asked, in a lesson couple months ago, ‘How about if the given one is really an identity? Can we use some values of x to check?’
‘Yes, of course.’I replied quickly.
is it an identity?
We, of course, can expand the and then judge ‘by naked eyes’ that whether it is an identity or not.
However, we can tackle the problem by the following way:
Put , , , so
Put , , , so
Put , , , so
Then, we can draw the conclusion that
is an identity and we may write .
Urm, just consider that
Observe that, the above a polynomial equation of degree at most 2 (well, in this case, the degree is 1 actually),
(i.e. it may be a quadratic equation or a linear equation etc.)
and, from above, we see that
are roots of that polynomial equation.
That is, a quadratic equation (i.e. a polynomial equation of degree TWO) has THREE DISTINCT roots (0,1 and -1), strange?
Be more abstract,
if the following equation has THREE DISTINCT roots
then, the equation should not be a quadratic equation (since a quadratic equation has at most TWO DISTINCT roots), so what is it?
Actually, it is not just an equation, it is an identity! It is the zero polynomial, that is
(see, it has infinitely many ‘roots’)
or we may write
For example, it is not necessary to expand the following
for showing it is an identity, just check that
there are more than 4 distinct roots to the equation. (Student, check it out.)
In general, if a degree n polynomial equation has more than n DISTINCT roots, then the polynomial is the zero polynomial.
Up to now, junior form students should laugh at my stupidity of using lengthy presentation for that trivial stuff, I guess.
But, this trick will be found in Pure Mathematics exercises, let me post some in random.
Suppose are distinct numbers, is the following an identity? Prove or disprove your assertion.
At a glance, it is a polynomial equation of degree at most TWO.
Hence a polynomial equation of degree at most TWO has THREE distinct roots ,
the equation is actually an identity!
Since we can substitute any value of for identity, say , we can create a question like
find the value of
For distinct numbers , prove that
Easy to know that are the roots of it, hence it is an identity.
Then we can create a not-that-trivial equation like
find the value(s) of
Well, follow similar way of thinking, we may create something involve , boring…
Suppose are distinct numbers such that
This is an old question, the trick is considering
Note that, this is a polynomial of degree at most , but there are distinct zeros*.
The following are very boring related posts, just for your debugging:
Many days ago, I saw a very beautiful picture on web:
it is something involves roots of polynomials, read the following for detailed description if you want to:
*Zeros of a polynomial means the roots of .