the question is
if g(x) is a polynomial satisfying
g(x)g(y)=g(x)+g(y)+g(xy) - 2
for all real x and y, and g(2)=5
then find g(3).
the answer is 10
please tell me how to do this ques.
my sir told me this could be solved in a quarter of a page
what you did is only correct if g(1) is not 2.
g(1) is 2 which can be calculated very easily
and besides, if this was true,
then,
g(n)g(1)=g(3)+g(1)+g(3)-2
g(n)(g(1)-2)=g(1)-2
g(n)=1(according to you)
which means for any value of n, g(n) will be 1, which is not possible.
for finding g(1)
g(1)g(2)=g(1)+g(2)+g(2)-2
5g(1)=g(1)+8
g(1)=2
g(x) is a polynomial, so
where N may be infinite. So
implies:
We can find the coefficients by matching powers of x and y.
Let's look at the coefficient.
Thus
Thus
or .
Now
Thus
The first tells us that , and the second tells us or .
Now
This gives:
The first tells us nothing new. The second says or . The third implies that at least one of or .
There are no contradictions so far, but it's getting really messy. The good news is that, in general, I'm getting that and some set of or for n > 0. Since we need we must have .
This fits with both our and . So I would say that
as we expected.
-Dan
thanks for the answer
you can get a more decisive answer by considering only n as 0, 1 and a general n and putting x or y=2
but basically using your solution
i never thought of doing questions tthis way, thanks
but i'm looking for a shorter solution if someone can help