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
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
$\displaystyle g(1) = 2 $ not so easily.
$\displaystyle g(1)g(1) = g(1) + g(1) + g(1 \cdot 1) - 2$
$\displaystyle g^2(1) = 3g(1) - 2$
$\displaystyle g^2(1) - 3g(1) + 2 = 0$
$\displaystyle (g(1) - 1)(g(1) - 2) = 0$
So $\displaystyle g(1) = 1$ or $\displaystyle g(1) = 2$. If $\displaystyle g(1) = 1$ then
$\displaystyle g(n)g(1) = g(n) + g(1) + g(n \cdot 1) - 2$
$\displaystyle g(n) = 2g(n) - 2$
Thus $\displaystyle g(n) = 1$ for all n. But the problem states $\displaystyle g(2) = 5$, so this cannot be.
Thus $\displaystyle g(1) = 2$.
-Dan
g(x) is a polynomial, so
$\displaystyle g(x) = \sum_{n = 0}^N a_nx^n$ where N may be infinite. So
$\displaystyle g(x)g(y) = g(x) + g(y) + g(xy) - 2$
implies:
$\displaystyle \sum_{m, n = 0}^Na_ma_nx^my^n = \sum_{n = 0}^N a_n(x^n + y^n + x^ny^n) - 2$
$\displaystyle a_0^2 + a_0a_1x + a_1a_0y + a_1^2xy + ... = 3a_0 + a_1(x + y + xy) + ... - 2$
We can find the coefficients by matching powers of x and y.
Let's look at the $\displaystyle x^0$ coefficient.
$\displaystyle a_0a_0 = 3a_0 - 2$
Thus
$\displaystyle a_0^2 - 3a_0 + 2 = 0$
$\displaystyle (a_0 - 1)(a_0 - 2) = 0$
Thus
$\displaystyle a_0 = 1$ or $\displaystyle a_0 = 2$.
Now
$\displaystyle a_0a_1x + a_0a_1y + a_1^2xy = a_1(x + y + xy)$
Thus
$\displaystyle a_0a_1 = a_1$
$\displaystyle a_1^2 = a_1$
The first tells us that $\displaystyle a_0 = 1$, and the second tells us $\displaystyle a_1 = 1$ or $\displaystyle a_1 = 0$.
Now
$\displaystyle [a_0a_2 x^2 + a_0a_2 y^2 + a_2^2x^2y^2] + [a_1a_2 xy^2 + a_1a_2 x^2y] = [a_2(x^2 + y^2 + x^2y^2)] $
This gives:
$\displaystyle a_0a_2 = a_2$
$\displaystyle a_2^2 = a_2$
$\displaystyle a_1a_2 = 0$
The first tells us nothing new. The second says $\displaystyle a_2 = 1$ or $\displaystyle a_2 = 0$. The third implies that at least one of $\displaystyle a_1 = 0$ or $\displaystyle a_2 = 0$.
There are no contradictions so far, but it's getting really messy. The good news is that, in general, I'm getting that $\displaystyle a_0 = 1$ and some set of $\displaystyle a_n = 1$ or $\displaystyle a_n = 0$ for n > 0. Since we need $\displaystyle g(2) = 5$ we must have $\displaystyle g(x) = 1 + 1 \cdot x^2$.
This fits with both our $\displaystyle g(1) = 2$ and $\displaystyle g(2) = 5$. So I would say that
$\displaystyle g(3) = 1 + 3^2 = 10$ 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