# Homework Help: Find the value of f(2)

1. May 10, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
If f(x) is a polynomial function satisfying 2+f(x)f(y)=f(x)+f(y)+f(xy), x,y belongs to R and if f(2)=5, then find the value of f(f(2))

2. Relevant equations

3. The attempt at a solution
The question clearly seeks the value of f(5). I put x=0 and y=2. Then
2+f(0)f(2)=f(0)+f(2)+f(0)
2+5f(0)=2f(0)+5
f(0)=1

Now I put x=0 and y=5
2+f(0)f(5)=f(0)+f(5)+f(0)
f(5)=f(5)

2. May 11, 2013

### ehild

f(x) is a polynomial function. What does it mean?

ehild

3. May 11, 2013

### haruspex

Consider a substitution f(x) = g(x) + c. Can you find a value of c that simplifies the equation?

4. May 11, 2013

### utkarshakash

Nothing special I can think of.

5. May 11, 2013

### Office_Shredder

Staff Emeritus
I'm going to second this suggestion

6. May 11, 2013

### ehild

f(x) is a polynomial function, of form f(x)=a0+a1x+a2x2+a3x3+...

What relations do you get for the coefficients from the given equation and data?

ehild

Last edited: May 11, 2013
7. May 11, 2013

### utkarshakash

I still don't know the degree of polynomial

8. May 11, 2013

### utkarshakash

Ok following your method I arrive at this

2+g(x)g(y)+(c-1){g(x)+g(y)}=3c-c2+g(xy)

9. May 12, 2013

### haruspex

Right, so what value of c will simplify that greatly?

10. May 12, 2013

### ehild

That is you need to figure out. From the condition f(2)=5 you get a relation between f(y) and f(2y), and that can be fulfilled with polynomials of a certain degree.

ehild

Last edited: May 12, 2013
11. May 12, 2013

### utkarshakash

The only number I can think of is 1

12. May 12, 2013

### haruspex

Right, so what equation do you get for g()? When you have that, suppose α is a root of g(x). What other root(s) can you then deduce?

Last edited: May 12, 2013
13. May 12, 2013

### ehild

I show my way as I think it is quite simple and straightforward.

2+f(x)f(y)=f(x)+f(y)+f(xy)

From f(2)=5 follows: 2+5f(x)=5+f(x)+f(2x) ----> -3+4f(x)=f(2x)*

f(x) is a polynomial f(x)=a0+a1x+a2x2+....+akxk+...

Plug into * and compare the coefficients of powers of x on both sides

-3+4(a0+a1x+a2x2+....+akxk+...)=a0+2a1x+4a2x2+....+2kakxk+...

-3+4a0=a0
4a1=2a1
4a2=4a2
.
.
.
4ak=2kak

What is the degree of the polynomial?

ehild

14. May 12, 2013

### haruspex

Ah, but mine is so elegant

15. May 12, 2013

### Dick

Elegant and cute, but less obvious.

16. May 12, 2013

### ehild

I must be dumb but still do not know your solution.

ehild

17. May 12, 2013

### Dick

I saw it. g(x)g(y)=g(xy) means if x is root of g then xy must be a root of g for ANY y. Severely limits the choice of roots. I think this is little too subtle.

18. May 12, 2013

### ehild

I reached here, but what after? f(x)=1+xh(x). But it is obvious as f(x) is polynomial, and f(0)=1 (obtained by the OP already). Find the possible root of h?

ehild

Last edited: May 13, 2013
19. May 13, 2013

### Office_Shredder

Staff Emeritus
I'm not sure what h is supposed to be here. But once you get to the post you quoted you say "ah, g(x) must be xk" and use f(2) = 5 to figure out what k is

20. May 13, 2013

### ehild

I do not see that "ah" Perhaps I stick to my version too much which gives the degree at once.

ehild

Last edited: May 13, 2013